Inverse functions provide a way to get some, but not in general all, solutions to equations that involve functions which are more complicated than polynomials.
Solve[Sin[x]==a, x, InverseFunctions->True] gives a single solution in terms of ArcSin. In fact, there is an infinite number of solutions to the equation, differing by arbitrary multiples of . Solve gives only one of these solutions.
When there are several simultaneous equations to be solved in terms of inverse functions, Solve may fail to find any solutions, even when one exists.
When inverse functions are allowed, Solve solves for first, then applies InverseFunction[f] to the result, equates it to expr, and continues trying to solve for the remainder of the variables.