Okay, so we need some background, so lets first get something out of the way
How currencies work
First, all currencies have no intrinsic value and merely represent standardization of value between two objects. In a barter system, I might sell you apples for oranges but we have a disagreement on the value of the two fruit because we are literally and figuratively comparing apples to oranges, which are different things that don't have a lot of common points between them. So, we use currency as means to exchange value. If I know how much it costs me to make one apple in a measure of currency to apples, and you know how much it costs to make an orange in a measure of currency, we can exchange the currencies for the fruits rather than the apple for the orange
So how do we measure the value of a currency? Well, the issuing government makes currencies to a set value. Currencies value are fixed or "pegged" to another currency or commodity. This means that a nation's currency (lets say Australia's fictional currency Dollary-doo ($D)) to another stronger currency (like the United States Dollar ($)) for this discussion) will try to achieve a ratio of their currency to their peg. In the case of commodities, the same principal applies, but instead of a currency, they peg the value to a thing that people buy on the world market and has intrinsic value and cannot be mass produced. Precious metals, such as gold, or energy commodities, such as oil, are often used for these values. Here the currency is "backed" by the commodity, and the nation uses a "[commodity] standard". With those terms defined, lets show how this works with something with a similar small value, a single purchase from McDonald's Dollar Menu.
Lets say that Australia's dollary-doo is pegged to the USD at a reate of 2 dollary-doos to One USD (2$D:$1) and $1 dollar can purchase one item on the Dollar Menu at McDonalds. This meas that in Australia McDonald's will charge 2$D for the same item American's buy for $1. Here, we have a currency peg to a Dollar Menu backed currency. If McDonald's changes the price of a Dollar Menu item to $1.20 because market forces need them to change the price, then the USD loses value and because Australia Dollary-Doo is fixed to the USD, it too loses value, now costing 2.40$D to buy a dollar menu item. To fix this, the U.S.'s Central Bank (The Federal Reserve) will start selling it's reserve of Dollar Menu items, thus bringing the item back down to the price of $1. This flood of dollars fixes the problem in the U.S. but in Australia, the Dollar-doo is still loosing value... it's not a common value to the Burger, but the Dollar, remember... purchasing Burgars is still costing them 2.40$D, but there's now more burgers and less dollars in the world... so Australia's Central Bank (the Bankery-Doo? shrugs) will start releasing more dollars for less Dollary-Doos. And other countries around the world will similarly adjust their exchange rates for buying back or selling more of their currency and commodities to keep as close to their set ratios as possible.
Now, that's how exchange rates mostly work... there are a few things that further complicate the matter. For example, the USD hasn't been a fixed currency since the mid 1970s, when it left the gold-standard, and is now a "Float" or "Fiat" Currency. This means that the value of the dollar literally represents the "Full faith and credit of the United States Government." If you'r not a fan of the United States government, this may seem like the dollar has no value at all, but what it means is that the United States economy is so strong, the dollar has value because you have to do buisness with the United States either directly (through trade) or indirectly (by doing buisness with nations who in turn will do buisness with the United State). Think of it as rather than the Dollar being on a Burger Standard, we instead peg the value of the Burger to the Dollar (and yes, there is a real life Big Mac Index that's used to show the price of a Big Mac in various countries). It's also helped that the USD isn't the only floating currency in town (though it is a popular one) and other currencies are tied to strong economies that let their economic might speak for their currencies' value (some other currencies include the Euro (EU), The Pound Sterling (UK), the Yen (Japan), the Rupee (India), and the Australian Dollar (not the Dollary-Doo, no one has faith in that). You might decide you don't like the United States, and peg to the Euro, but the Euro economy and the USD economy do interact frequently.
The other issue is that you don't have to peg to one currency, and can instead have a "basket" currency, which means you'r pegging to two or more currencies and trying to keep yours in balance with both of those (sorta demonstrated by the Dollary-doo to Burger example).
Suffice to say that currency exchange has a lot of extra details that will overcomplicate what you need to understand about a currency used by any nation and requires math and economics theories that you could write books about. I am not doing that here (partially because I'm not sure I understand all of it). What you need to know is that in a vacuum, currency has no intrinsic value. A bank note is just a piece of paper and a coin is just a disk of metal. It's value comes from setting it to another thing that does have value. This is usually your own economy, another currency, or another item on the market. Supply and demand change the values of things and thus the value of one currency to another.
Back to the Question
So why was that important? Because as a Float Currency only works if the issuing government can guarentee it. Currency's value is governed, like any other commodity, by supply (how much is there available to buyers) and demand (how much people want it). Too much supply and there will be no demand which causes inflation (Too many people have too many dollars, but the amount of burgers are running out means the Dollar cannot buy as many Burgers and the Burger price rises) while not enough supply but plenty of demand means deflation (there aren't enough dollars to buy all the burgers, so the Burgar price falls). Governments are then motivated to keep their currencies competitive with other nations and will carefully look at the markets and release currency for currency on ratios they want to control for and avoid going full Wiemar Republic (never go full Wiemar Republic). If they have to issue new currencies, they do so only in limited supplies based on market forces, thus avoiding inflation or deflation... not just at home but aboard. Both ways in excess can be devastating. And if a big engine of world economics slows down, that's an international economy issue, not a national one.
One way of control is that governments will only accept their legal tender. If someone is counterfiting U.S. dollars in a foerign economy, the U.S. can't arrest them... but that money will find it's way into the exchange of currencies... and the U.S. Fed won't buy back currency it did not issue... thus the counterfit is worthless to the Central Bank that holds it (because the value is based on the U.S. saying it's real money and accepting it in transactions). So the central bank of a foreign nation won't accept the fake bill because it's the same as giving them monopoly money... they can't use it. So it's not worth giving their legal tender in the same amount for your fake bill. You can try to buy a burger with it, but the burger puts that money in the bank too... so they want nothing to do with it.
In fact, if it becomes a big problem for your nation, the currency issuer will just stop buying your currency, which, is bad because you can't adjust your exchange rate if your a peg, and floating currency requires you to have an economy that we can trust is good... another floating currency calling you a damn dirty counterfitter will cause other nations to lose faith that the currency they buy from you is valuable. Now... without the ability to compete with other exchanges, your dollars and cents are now paper and pogs... and it could be worse... paper and pogs might be worth more than than your currency. You've just gone full Wiemar Republic (Never go full Wiemar Republic) (and yes, the currency of the Weimar Republic (The Deutsch Mark) was so worthless, that one story tells of how a woman took a wheelbarrow full of Marks to the Market to buy bread and while shopping, left the cash out of sight... she turned around and was horrified to find a thief had robbed her... to the tune of wheelbarrow... the bills were left lying in the street, as the thief dumped it out to make a speedy get away.).
By the way, this is one of many reasons why North Korea's economy is so poor. I'm fuzy on the nature of the events, but North Korea was able to get it's hands on the equipment used by the United States to make USD and tried to flood the market with illegal "super-counterfit bills" that were so much like the real deal, the Banks couldn't tell the difference. The U.S. responded that they wouldn't take any cash that passed into or out of North Korea... and since one has the largest share of economic wealth in the world, and the other's currency was so worthless that, for a time, snack-cakes replaced it as currency (I am not making this up, a freaking moon pie was a more reliable medium of trade in North Korea for a brief period of time), and it didn't take long for the world's markets to make the decision as to which economy they were not gonna trade with. Even China, North Korea's most economically powerful friend, still wanted the USD more than the moon pie.
Similarly, the Confederate States of America's economy tanked and in part lead to their loss in the U.S. Civil War. They relied on Cotten making them invaluable to foreign markets. However, they couldn't leave port due to bloc aids by the Union and Demand skyrocketed, making the limited supply of other content producing nations much more valuable, or less expensive then getting it from the South, as the foreign markets had to pay a higher price, both economically and morally. Those who didn't care how the cotton was produced, so long as it was cheap could no longer buy it for cheap and those who cared about how cotten was produce didn't want to buy cheap if it meant condoning slavery. The south, desperate for cash to fund the war, saw prices of things increase both because the couldn't import cheap goods, and their currency wasn't able to keep up with market changes. It also didn't help that the printing of Confederate Dollars was done on the cheap meaning that the Union could flood the market with counterfeits, further ruining the economy, to such a degree that their counterfeiting was often detected because the fake bills looked realer than the real bills. Hard to have "Full faith and credit" in the Confederate army when your currency looks like it was counterfeited and people use the realistic stuff... which is "worthless". The economic impact was so bad that the states that made the Confederacy are still suffering from the damage to this day, 158 years after the Confederacy ceased existence.
Russia is also still realing from their bad exchange rate of precious metals to Rupals that caused an economic nightmare in World War I! Their exchange rate was 1 Rupal : 1 unit of precious metal, so when the economy started to falter, the started using their metal reserves to buy back Rupals... which if you don't see a problem with that, you give me one dollar in USD and I'll give you a dollar in Monopoly Money and then you go buy me something off the McDonald's Dollar Menu and I'll give you a dollar in exchange for that purchase... and if you're still lost, Russia buying the monopoly money.
In both cases, "Mo' Money, Mo' Problems" is certainly true when the money is losing value... but everything else isn't. And the realy danger is that a few days under a bad economic choice can take decades of recovery if you're really really good about fixing the problem... and to outsiders looking in, your claim of economic acumen of that caliber will only remind us of just how we got into this mess in the first place. Tsarist Russia collapsed because of their financial mismanagement... Several Confederate States were ready to succeed from the Confederacy and only stopped when the Union won the war and solved the confedercy's economic woes in a fashion that is akin to how vets solve the issue of a broken leg for a horse.
Conclusion
Zimbabwe and Syria might not like the United States Government... but, like everyone else in the global economy there own currency has no value unless it's fixed to the value of other things and they will likely peg it to another nation's currency and exchange all sorts currencies, not just their own, to keep the two currencies close in value and this means their money will somewhere in the exchange market, will get a valuation in USD. If the US isn't going to take currencies that are fake, it's your interest not to take worthless paper that looks like real dollar bills. You don't have to enforce U.S. laws against counterfitting and alert the United States, but you're not helping yourself if you buy too much fake at the same rate as real currency, even if the fake currency isn't your own. You're still paying too much for it's real value and thus, inflating your currency faster than another economy. While you're free to hate the United States, most nations agree with some of the U.S. economic policy, specifically the part best summed up as "We want those dolla dolla bills, y'all".
