Start with q, a linear combination of n dimensional simplexes in S. By assumption, q is a cycle, with boundary 0. In one dimension, you might picture q as a set of directed arcs, such that the in-degree of each vertex equals its out-degree.
For each simplex v in q, build a new n+1 simplex u as follows. Map e0 through en as directed by v, and map en+1 to w. Fill in the rest of the simplex linearly. Since lines join w to all of v, there is no trouble. If you're in the mood for technical details, you can prove u is a continuous map from δn+1 into S.
Do this for each v in q, and apply the same coefficients, giving a linear combination of n+1 dimensional simplexes that I will call r.
Let f be a face of a simplex in q. In our one dimensional example, f is a vertex in the digraph. It appears +k times and -k times in the boundary of q. In each case the cone from w to f is an n dimensional simplex in the boundary of r. It appears +k times and -k times as we remove the various vertices of each simplex v in q. The faces incident to w all cancel, and the boundary of r equals ±q. In other words, q is in the image of Gn+1, and hn is trivial.
Many of the spaces you know, such as simplexes, n dimensional balls, real space, and generalized euclidean space, are star convex, and have trivial homology groups across all positive dimensions. This generalizes to any space that is homeomorphic to a star convex set.