In $\S$ 9.3 of the book "Mirror symmetry" (Vafa, Zaslow eds.) the authors formulate the following general localization principle for computation of integrals with respect to both even and odd variables: Assume there is some supersymmetry transformation of variables. Then the integral becomes localized on the field configurations for which the fermionic variables are invariant under the supersymmetry.
I would like to understand the precise meaning of this sentence. Though several non-trivial and beautiful applications are discussed in the book, I do not understand how they are deduced from this principle due to the fact that the principle itself seems to be too vague. The explanation on pp.158-159 how it works is too concise for me, and it is not clear how that example can be generalized to other situations.
Question: how to formulate a precise statement behind this principle (the situation of finite dimensional integrals would be enough for me for the moment)? Can I read a proof of it somewhere?