Hamilton defined the Ricci flow in 1982 to prove the famous three-dimensional sphere theorem. How singularities of solutions of the Ricci flow could identify the topological data predicted by William Thurston's geometrization conjecture?
How did Thurston's geometrization conjecture eventually lead to the proof of the Poincaré conjecture?
Answer to your second question: Thurston's geometrization conjecture states that there are only eight geometries in closed 3-manifolds. Only one of these geometries is a homotopy sphere. Therefore any homotopy spere must have the geometry of the 3-sphere and thus there cannot be false 3-spheres, which is the Poincare Conjecture.
Answer to your first question: The 2-manifolds on which the Ricci flows are defined are in fact Morse surfaces on the closed 3-manifold. The singularities of the Ricci flow are critical points of the Morse function and they define the manifold up to homotopy and homology, which is sufficient to identify the Thurston geometry. So, a Ricci flow describes Morse surfaces with the Ricci tensor and makes it possible to calculate something with differential equations, instead of the traditional way of doing 3-manifold topology with geometric topology methods, like surgery or triangulazation, that necessarily are rather hazy and Poincare Conjecture proofs using these old methods were never accepted as they were not clear enough.
Hope this helps, I cannot remember much more from 40 years ago when I was on that field.
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