Perelman Again (Tao's Version)

One can read in Professor Tao's Lectures the following:

[Geometrically, positive scalar curvature means that infinitesimal balls have slightly less volume than in the Euclidean case; positive Ricci curvature means that infinitesimal sectors have slightly less volume than in the Euclidean case; and positive sectional curvature means that all infinitesimally geodesic two-dimensional surfaces have positive mean curvature. I don't know of a geometrically simple way to describe positive Riemann curvature.]

Terence Tao 

I am trying to understand the "big deal" in Poincarés Conjecture, now Perelman- Hamilton's Theorem. Maybe Tao is a good place to start.