Let T be the tangent vector( velocity) of γ, then dT/ds is the curvature vector( acceleration), and the norm of its projection onto Π is the geodesic curvature. Then the statement is a consequence of the following statement:

Projection of the acceleration of a curve equals to the acceleration of its projection.

This is easy to see if your choose a coordinate system so that the projection is (x,y,z) |->(x,y), so the acceleration is (x'', y'', z''), its projection is (x'',y''). And the projection of the curve is (x,y), its acceleration is (x'',y''). DONE.

The deep idea is, the projection onto Π is an isometry in a infinisimal neighborhood of p.