붙다 is one thing sticking to another (이거 너무 오래 둬서 붙었네), and two of its idiomatic uses show up here. It means passing a test or audition: 아이돌은 오디션 봐서 붙어야 되고, 나는 거기서 은상 받아서 붙었고~. And it means squaring off against someone: 어디서 구슬을 가져와서 붙자고 온 거야. It also just means staying stuck to a place (월세 지원해서 붙어 있을 거기 때문에). 붙이다 is you sticking something on; 붙다 is it sticking or catching.