Legendre presented an ingenious proof in 1794, 2 the first step of which was to project the polyhedron onto a sphere. The specific way in which he carried out this projection (described in Ex. 26) required him to assume that the polyhedron was convex, just as Cauchy would also do, 20 years later. But, as we have noted in the context of Cauchy’s proof, convexity cannot actually be relevant to a topological result. Therefore, once again, we shall sidestep this historical artifact and present a more blatantly topological version of the argument—one that does not hinge on convexity, but that stays true to Legendre’s essential insight.
I am a bit confused about how one can show the result with projection because when we project, we collapse a bit of information. In particular, the projected figure on the surface of sphere need does not really have a relation with the Euler Characteristic with the that of the original figure we begun with because many figure give same projection. (Well, once we know proof, then we know it's always 2 but we don't know that yet)
So, how does the proof through projection deal with this issue?
Later the book says this:
As before, imagine our polyhedron P to be a curved, topologically spherical, rubber membrane, with dots (vertices) and connecting curves (edges) drawn simply on its surface. Instead of collapsing the polyhedron, as we did in Cauchy’s proof, let us this time inflate it like a balloon! We thereby arrive at a polygonal net covering the surface of an ultimately spherical balloon. Note that the edges that result from this are not geodesics, but, as we now explain, we can easily make them so without altering the Euler characteristic.
How does inflating the concave figure relate in any way to projections? I don't see how this connects to the previous idea at all.