You write “As I understand, it wouldn't make sense for an individual or company to make new loans every year to pay old ones” but that's incomplete in many ways.
It's indeed a bad idea for an individual to make new loans to pay old ones. That's because they have a finite lifetime, limited use for the money (mostly, individuals build up debt to move consumption in time or simply because they can't meet obligations but they will have to get money through other means to repay it) and because they would quickly have deteriorating credit and pay a lot of interests.
Companies by contrast certainly can do that profitably, as long as they can find investments that are more profitable than the cost of the debt. And, as long as there is a healthy financial market and the debt remains manageable, there are actually accounting reasons why it's a bad idea for a company to be entirely debt-free.
States are in an even better position to borrow money. They exist essentially indefinitely and provide very strong guarantees because their future income stream (i.e. taxes) is even less likely to disappear than a company's. That's why many (but by no means all) states can currently finance themselves at very low interest rates, which incidentally shows that a debt of 50 or even 100% or more of GDP isn't actually that “large” compared to a state's future ability to pay.
(Note that public debt is conventionally expressed as a percentage of GDP because the absolute value would just reflect the size of the country but there is nothing “magical” about 100% or any other threshold, it's just a way to scale the number. Some states have or had at some point in history much larger debts than that.)
Importantly, a state, especially in a large and moderately open country (but not in a tiny wide open economy like say Luxembourg), is also in a very different position than an individual or a company because it can actually influence the whole economy. When an individual spends less, there is no reason why it would impact their income and they simply end up with more money, which can be used to reduce debt.
Not so with a state. When it spends less, it means that it pays less money in wages, subsidies, etc. and a state is big enough that it could depress an entire economy and ultimately damage the state's own revenue (i.e. taxes). Conversely, it means that if the economy is depressed, spending more can jumpstart activity and result in increased tax revenues which both reduces the deficit and makes the total debt lower relative to a growing economy. The feedback effect is so strong that you can't think correctly about public debt through analogies with personal debt.
This is important because it means that in some conditions (I am certainly not suggesting states can always borrow and spend arbitrary amounts of money), building more debt and spending the money can profit everybody and the state itself, even if it's used for something that sounds completely unproductive (Keynes' proverbial burying of old notes to let people dig them up again). It's not even necessary for the debt to fund things that will increase productivity and ultimately create wealth in their own right like infrastructure projects.
Finally, debt issued by the state has some useful functions in the financial sector. Banks and other investors currently have a lot of money which they don't know how to use and there are technical reasons why it can be useful to offer them something to do with it. That's another, even less intuitive reason for states to borrow money.
Of course, that does not mean that deficit spending or, conversely, attempts at reducing the debt through reduced spending are motivated by a clear understanding of all this. Ulterior motives and political expediency obviously play a big role and not only on the spending side of the equation.
Also, technically, countries issue new debt because they need the money to meet obligations (pay something or roll over some old debt). Of course, they often need to borrow money because the budget is not balanced but that's not something you can control that easily.
That's an important distinction because states cannot directly act on the budget or simply decide that they don't want more debt. What they can do is make decisions on spending and taxes but the resulting budget outlook also depends on interest rates and economic growth (and the state's own ability to carry out its decisions, which is not always trivial). An extreme example of that would be Greece in recent years. It performed massive cuts with an explicit goal of balancing the budget and reducing debt but it ended up needing even more money.