Intuition for definition of projective module

An equivalent characterization of projective modules are those modules $P$ for which any short exact sequence $0 \to A \to B \to P \to 0$ splits; that is, $B \cong A \oplus P$. In this sense, $P$ generalizes free modules (which share this property), and so this is even further a generalization of vector spaces.

This viewpoint is also suggestive of the terminology - $P$ can be viewed as a direct summand of any module from which it admits a surjective homomorphism, so $P$ is in fact a projection of that module onto one of its direct summands. In particular, since any module is the image of a free module, $F\cong K\oplus P$ for some free module $F$, so $P$ can be viewed as a projection of a free module.

The most intuitive way to think about projective module(s) is probably direct summand(s) of some free module, so in a sense it is a generalisation of "complemented subspace" in the study of Banach spaces, except you couldn't really make that generalisation since, e.g., $c_0$ is not complemented in $\ell^\infty$ despite both being Banach.