I'm trying to understand the argument given on the wiki page for complex projective space that $\mathbb{C}P^n$ diffeomorphic to $S^{2n+1}/U(1)$. I don't understand why the last line. Is there some standard theorem about quotients of compact lie groups?
Not quotients of Lie groups (the sphere is in general not a Lie group). But the quotient of a manifold by a free and proper group action, yes. If the Lie group $G$ acts on the manifold $M$ on the right, then the general criterion you need to get $M/G$ to be a smooth manifold (so that the canonical projection $\pi\colon M\to M/G$) is a submersion is that the relation given by the action, $R = \{(x,y)\in M\times M: y = x \cdot g \text{ for some } g\in G\}$, be a closed submanifold.