In short, they're fun. ;-)
If you love math, going to a math camp is something that you should definitely experience at some point in your high school. These camps provide exposure by allowing you to meet a diverse groups of students who are united by their passion for mathematics and computer science. The math camps I have attended have probably been some of the best parts of my life, and they introduced me to a wide range of people who are now some of my closest friends!
On another note, you also get to learn various super cool math concepts from renowned professors, and get an introduction into a variety of concepts in higher mathematics.Â
Indian IMO training camp - Well, you can't "apply" to this, but I just wanted to highlight that attending this is a huge experience - so it is definitely worth it, even if you think you're not likely to make it to the team. To attend this camp, you'll have to clear INMO (check out more details here)
Math Beyond Limits - I attended the MBL Balkans 2023, and it was an amazing experience. I got to learn a lot of very cool math, and I also had the opportunity of meeting people from various parts of the world who were united by their love for math. It was a super enjoyable experience, and we also had a ton of fun other than the math, and I'd definitely recommend anyone reading this to apply!
STEMS, CMI - I attended this earlier this year, and while it is usually a short camp, it is still super fun! To attend this, you'll have to clear an olympiad conducted by CMI (more details here)
ESPR, WARP, SPARC, Atlas - These are some super cool "applied rationality" programs, that while revolving around math also help you understand a ton of other fields and how all of them connect together. While I haven't attended any of these myself, I think they're all really cool and worth applying too!
Other US camps (links here)