Nothing much is going on. There are two simple successor-like operations on infinite cardinals. One is exponentiation: from $\kappa$ you get $2^\kappa$, but you can’t say much more about its size than that it’s bigger than $\kappa$. The other is the actual successor operation: from $\kappa$ you get $\kappa^+$, which is the smallest ordinal that admits no bijection onto $\kappa$. This is the real successor to $\kappa$: it’s the smallest cardinal strictly bigger than $\kappa$. The generalized continuum hypothesis is the assertion that $2^\kappa=\kappa^+$ for all infinite $\kappa$, i.e., that these two successor-like functions are really the same function; as you say, it is independent of ZFC. The axiom of choice enters the picture when you want to prove that the cardinal $2^\kappa$ is well-orderable; without AC you can’t guarantee this. The successor $\kappa^+$, in contrast, is always well-ordered.