Strictly speaking the prerequisites are relatively low. It isn't necessary to know much past the basic Riemannian geometry of curvature and geodesics (but to know them well). Anything more advanced, such as the Toponogov theorem, can be picked up as needed. In particular, there are no sophisticated algebraic structures involved, and there's only really any topology at one final point, outside the main line of the argument, and it's on the level of an exercise in Hatcher.
Having said that, the proof is entirely PDE-based and so understanding it requires a strong appreciation for PDE estimates and for inequalities. This is a nontrivial thing to acquire (think of all the "pure" mathematicians who scoff at what they see as applied), but it doesn't require knowing too many technical facts.
At any rate, the starting point for anyone would be a Riemannian geometry class and Hamilton's 1982 paper "Three-manifolds with positive Ricci curvature"