Product of paths is a very geometric operation - it is the concatenation of the paths.
I'm trying to phrase the analogue for the addition operation of chains of n-simplices (and of the induced operation on homology classes). Is there such a thing? Did I miss anything obvious?
Many thanks!
You did not miss anything: there's no particular geometric intuition, it's really just algebraic formalities. A chain is just a "vector" assigning a numerical coordinate to each singular simplex, with all but finitely many coordinates equal to zero; addition of chains is just vector addition, i.e. coordinate-by-coordinate addition.
That doesn't mean that there's no intuition at all. But to get good intuition I would advise learning about simplicial homology, with particular attention paid to examples of small, finite simplicial complexes. Even just studying the homology of small, finite 1-dimensional simplicial complexes a.k.a. graphs, is useful. On a graph, a 0-chain is just a little number floating next to each vertex, and a 1-chain is just little numbers floating next to each edge. If you have one blue $1$-chain (little blue numbers floating next to each edge) and one red $1$ chain (little red numbers floating next to each edge) then their sum is a purple $1$-chain (add the little blue and little red number floaing next to each edge, to get a little purple number).
The real test comes as you now apply the boundary operator, and study cycles and boundaries and how they are used to compute homology groups. That's where you'll really learn to develop new intuition and new understanding.