QuickCheck et al do a random walk to find a value that fails the tests. Typically the value is large, complicated and unwieldy - particularly if the test involves nested structures. It might by 10s of KBytes in size, and most of the value is irrelevant.

Minimization is the process of starting from that known value and simplifying it as much as possible while still failing the test. For example QuickCheck discovers a 1235 element list, that fails. The problem happens to be a 0 in the last element. Minimisation might simplify that list to [1, 0]. So finding the cause is quite a bit easier.