What exactly is the impact of the hidden subgroup problem on cryptography?
As far as I understand, the HSP is a hard problem such that:


some types of HSP (namely those operating in an abelian group) can (theoretically) be solved efficiently on a quantum computer (assuming one can be built);
many types of public key cryptosystems can be reduced to the HSP: if you can solve the HSP you can break the key.


In particular, integer factorization and discrete logarithm (in any abelian group, which includes elliptic curves) can be reduced to HSP over abelian groups, and thus easily breakable on a quantum computer. This is not really new: Shor's algorithm already does that. It just happens that Shor's algorithm is a special case of HSP.

The interesting part is that lattice reduction can be reduced to HSP over a non-abelian group, for which no efficient algorithm is known yet. So this formalism is actually giving reasons on why lattice-based algorithm may survive in the post-quantum world.

In your summary of HSP, one might add two details: "finding H" really means "finding a set of elements of G which generate H" (i.e. H is the smallest subgroup of G which contains all those elements); and: "given f" means "a black box implementing f is given, and can be invoked repeatedly on any inputs".