Why is the kernel of an integral transform called kernel?

The kernel of an integral transform is called kernel with more right than the kernel (nullspace) of a linear transformation. The two meanings have absolutely nothing to do with each other.

An integral transform is a black box that takes some function $f$ as input and produces some (other) function $Tf$ as output. The kernel is what's inside that box. It is the "program" that encodes the exact manipulations to be applied to $f$.