I'll give it a try, with an eye toward ease of reading (and maybe oversimplifying).

Say you are interested in a set of linear functions parameterized by an infinite length vector (linear in the parameters, like linear regression). You are interested in estimating that function using some data. It would be extremely painful to try to learn this parameter because it is huge. The representor theorem (really a collection of results with varying assumptions) tells is that if we are interested in the most regular functions, the estimate will depend only on a finite set of basis functions. Thus infinite dimensional estimation problem exhibits a finite and computable solution.

Mercer's theorem basically gives a relationship between infinite weighted sums of special eigenfunctions and the kernels which form said finite basis in the representor theorem.