I know that most students list a lot of extracurriculars on their applications, which include location and school-specific things like club presidencies or community service participation. How can colleges verify that students aren't lying about these activities? Do they even look into it, or do they pretty much just take people at their word?
Admissions officers don't have time to investigate all of these smaller claims, but they will almost certainly do some research if you participated in a very prestigious program or won an impressive award. The greater impact the claim has on your potential as an applicant, the more likely it is that schools will do some fact-checking. It's never worth it to lie on your college application! It will derail your education down the line if discovered (you'll get kicked out or have your degree revoked).
If you're applying for a top school and the extracurricular is a spike (your most impressive activity on your application), they will almost certainly look into it.
For example, when I applied to Harvard, I had one activity (call it X) that stood out way above and beyond others -- it was an high international ranking in a specific area. Now X had it's own official web site where you can see my ranking -- so you'd think that'd be verification enough right? But actually I was told by my coaches for X that they were actually contacted by Harvard admissions the verify I actually did what I said! That's how intense they were -- they verify things even when there are official sites and listings shows off what you did!
To more directly answer your question, here's how they often do due diligence: - Check to see if there is an official website listing your name. - Calling your references to see if the most impressive things make sense. - Seeking out your coaches in these areas to verify.
Of course, they will NOT verify every little activity. If your most impressive thing is ranking as a state top 10 lacrosse player, they will verify that, but then if you have a second entry that's like president of chess team, they probably won't verify this.
However, the risk-reward calculation is usually against you. The things they don't verify wouldn't make a huge impact on your application anyway, and on the chance you're caught in the lie (you say you're president of chess team, your recommender says you came to chess team once), it will totally sink your application.
And the above is just solely the practical considerations.