A proof strategy was given by Richard Hamilton: imagine the manifold is equipped with a metric. Follow the Ricci flow of that metric through the space of metrics. As the flow proceeds along parameter time, it will from time to time pass through metrics that describe singular geometries where the compact metric manifold pinches off into separate manifolds. Follow the flow through these singularities and then continue the flow on each of the resulting components. If this process terminates in finite parameter time with the metric on each component stabilizing to that of the round 3-sphere, then the original manifold was a 3-sphere.
The hard technical part of this program is to show that the passage through the singularities can be controlled. This was finally shown in (Perelman 02).
See at Ricci flow for more.
Laurent Bessieres, Gerard Besson, Michel Boileau, Sylvain Maillot, Joan Porti, Geometrisation of 3-manifolds (pdf)
Notes from a survey talk: