Because in many parts of the US, the major universities were plunked down in a 10km by 10km square of wilderness, with literally nothing else around.

Sometimes a big town grew alongside.

Sometimes it didn't.

Given that universities are non-profits, and therefore are not required to pay any municipal property tax, there were many that simply could not be patrolled by the neighboring towns. So they had to hire their own police forces.

A university police force has police powers within a university campus--it can arrest people. I will add that in my college days, the university police did not carry firearms, though now many do.

I imagine that the custom arose because Anglo-American universities tended to be rural or at least in small towns, unlike the continental ones. Remember that universities until about 1900 were exclusively male, so that one was gathering a large number of young men at prime hell-raising age. A college of 500 students might generate more work for police than a town of 2000 citizens might otherwise have, or care to pay taxes to police.

The university that I went to has ~40,000 students between undergraduate and graduate students, and an additional ~20,000 faculty, staff and other people there at any given day. There were thousands of people living in student housing on campus and tons of sporting events that could draw crowds near the six figures.

They have a deputized police force of people available 24/7 for things as mundane as walking people home safely and as complicated as enforcing local/state/federal laws. They are sworn officers that have an agreement with the city they operate in that allows their jurisdiction to bleed into the city proper.

I never interacted with them when I was a student, but it's not out of this world to have a small police force for the campus - there were more people there on the average weekday than the city that I grew up in, which also had it's own police force.

Many/most laws in the US are selectively enforced. A university police force is a prudent way for many/most universities to have university matters taken care of in a way that suits the university. Typically this means doing a lot of menial things (e.g., dealing with underage drinking, esp. in public) that might create liability or a PR nightmare for the university that the local police might not be inclined to deal with. The university police can often be more likely to de-escalate certain matters than local police, and this is generally a good outcome for the school.

At wealthier universities, this is also a way to convince parents that their kids are safe. Harvard, Yale, Penn, and Stanford (off the top of my head) are all in or near not-so-safe neighborhoods -- although gentrification of those areas are making this less of an issue these days.

I'm European too! Can't speak to the exact ins and outs of why they have them and the history behind it, I just know that they do.

A big school in a small town could completely overwhelm the local police department. So schools and towns work together to have a campus police department.

https://en.wikipedia.org/wiki/Campus_police

Hiring off-duty police officers to provide security for events is very common. Without their own police departments, universities would be doing this constantly and would likely not be able to find sufficient numbers of off-duty officers for all events for which they wanted to provide security. It also allows universities to provide a higher concentration of law enforcement officials on their campuses than would otherwise be provided by the police departments in the surrounding municipalities. There are definitely problems with policing in the US, but notable exceptions notwithstanding, campus police tend to be among the more community-minded LEOs.