"Feynman Path Integral can be used to compute geometric invariants of a space."
There several different approaches doing this. Let me try to explain one of them, but remember it is not the only.
The point is that first you should omit the world "Feynman" !
Just integrals are useful to compute geometric invariants - for example Gauss-Bonnet theorem expresses the Euler characteristics as integral over manifold.
Word "Feynman" appears when we consider infinite-dimensional manifolds - so we need to "integrate" over infinite-dimensional spaces.
However we are NOT really interested in geometry of infinite-dimensional manifolds - we are interested in finite-dimensional manifolds.
It appears that in some situations infinite-dimensional manifolds are either contractable to finite-dim ones or there is some heuristics which relates invariants of infinite-dimensional manifolds and finite-dim. For example if you consider loop space of M, manifold itself is embeded into loops(M) as subset of constant loops. If you consider the rotations of loops - then constant loops are fixed-point of this action - so in this case manifold is inf-dim but fixed point set is finite-dim - so we considering equivariant calculations we can get the result on finite-dim results.
So the red-line is the following -
in finite-dim case you integrate closed forms on manifold - and get invariant
in Feynman setup certain integrals reminds closed forms on some inf-dim spaces (loop space or whatever) so integrating it you get invariant.
(In some situations "closed form" menas with respect to BRST differential).
The classical examples are related to Mathai-Quillen formalism and interpretation
in terms of QFT.
Let me suggest to look a
M. Blau The Mathai-Quillen Formalism and Topological Field Theory
And cite the abstract:
"These lecture notes give an introductory account of an approach to cohomological field theory due to Atiyah and Jeffrey which is based on the construction of Gaussian shaped Thom forms by Mathai and Quillen. Topics covered are: an explanation of the Mathai-Quillen formalism for finite dimensional vector bundles; the definition of regularized Euler numbers of infinite dimensional vector bundles; interpretation of supersymmetric quantum mechanics as the regularized Euler number of loop space; the Atiyah-Jeffrey interpretation of Donaldson theory; the construction of topological gauge theories from infinite dimensional vector bundles over spaces of connections."