Herd immunity* is one aspect.
Limiting mutations is another.
The Covid virus has shown so far a disturbing, and not totally expected, capacity to quickly evolve itself into more infectious variants (and possibly more lethal/harmful ones as well, if the impact of Delta on young people in the UK turns out to be actually worse). Any large scale population which the virus can reliably infect is essentially providing incubation services for further viral evolution.
This is why the "big, successful, mutations" are associated with countries that had either quite big populations (India) or had high rates of infection (Brazil, South Africa, UK - possibly due to being the first country to mass-sequence, India again).
Granted, denying those "incubation pools" to the virus in the USA will not avoid mutations arising elsewhere. But there will be only limited possibility of quarantine-by-travel-bans if a new mutation happens within any given country.
This is also why providing vaccines quickly to poorer countries is not just ethical altruistic behavior but also basic self-interest for rich countries.
Economic relief for businesses
Asides from places that have decided not to have restrictions at all, like South Dakota, most places have had to navigate between throttling infection rates with various operating restrictions and trying to limit damage to retail and service businesses.
Take my home province of BC, Canada. Our health authorities have walked a fine line between being too prudent for some and too permissive for others. We've not had any real hard stay-at-home orders, except for March-April 2020, but have had restrictions on restaurants, bars, cinemas, etc... and lots of businesses failed. Canada also didn't really have any high vaccine availability until March/April 2021. That time period coincided with a big spike, which most of the world experienced, for which we had to bring another, tighter, round of restrictions to keep rates from spiraling out of control.
I can tell you that everyone saw the needle start to move, quickly, once partial vaccination rates passed 40-50% in May/June. Now we are nearing 80%, with still about .1% extra first-shotters per day, doing their bit. It has made a huge difference and almost all restrictions on businesses and individual have now been lifted, without it seeming like much of a health risk tradeoff. That's a luxury that we are blessed to have, as a rich nation, and it seems inconceivable to waste it due to unconfirmed fears and rumors. What was it after all that a certain Lt. Governor of Texas said??? I want to live smart and see through this, but I don't want the whole country to be sacrificed.. Well, Dan, let's hear you say that again, for vaccinations this time.
* Herd immunity, via infectious exposure, is apparently not a very credible outcome with Covid. Manaus, in Brazil, had supposedly reached that late in 2020, with very high infection rates as evidenced by antibody tests. Nevertheless, they got whacked again in early 2021 when new variants reinfected people. To over-simplify: Covid seems more like the common cold rather than measles with regards to keeping natural immunity after infection. (re. vaccine-based herd immunity considerations, see @JJJ's answer)
p.p.s. Why is it so important to vaccinate so many?
If you get the sense of moving goalposts, from 60-70% of vaccination rate like Fauci said was needed earlier to 80-85% now, you're not wrong. But there is a reason for that. R0, the basic infection rate is about how many non-immunized people a contagious person can expect to infect even when precautions are taken. It's the inverse of the vaccine requirements. If a person can infect 3 people (R0 at 3, covid in early 2020, without much restrictions) then if 2 of those 3 people are vaccinated, the virus only really makes 1 person sick and the rate is constant, neither increasing nor decreasing. If R0 is 5-ish, like Delta is suspected to have mutated to, then the proportion of people needing to be vaccinated is now 80%, not 66%, so that at most 1 person out of 5 gives the virus a new home. Just because of the darn virus' increased transmission rates. That's why measles (R0 18-20) needs 95% vaccination rates. If on the other hand Delta's R0 is now 8-ish (the upper range of estimates for it), then you'd need 87.5% vaccination rates to keep it down. Less than measles, but considerably more than the initial estimates before variants showed up.
The high claims of heightened R0 may seem a little bit alarmist, but consider that, in quite a few areas, variants established themselves as the dominant strains within 3-4 months of their arrival, which indicates considerably better performance at infection, compared to their older siblings.