The evaporation increases because spacetime is bent more rapidly by smaller black holes than big ones. It’s the same reason you can enter a supermassive black hole without being spaghettified, because the curvature never reaches a point where there’s a huge difference between your feet and head, sort of thing.

That curvature drives the evaporation rate, because the particles flying off are virtual particles whos partner fell in to the black hole with it flying outward. That happens far less often when the curvature is so low that any given point around the black hole is almost flat spacetime.