I've read that abelian categories can naturally be enriched in $\mathbf{Ab}.$ How does this actually work? Wikipedia defines the notion of an abelian category as follows (link).

A category is abelian iff
  
  
*
  
*it has a zero object,
  
*it has all binary products and binary coproducts, and
  
*it has all kernels and cokernels.
  
*all monomorphisms and epimorphisms are normal.
  

It later explains that an abelian category can naturally be enriched in $\mathbf{Ab},$ as a result of the first three axioms above.
How does this actually work?

I would leave this as a comment, but it seems that I don't have enough reputation to do that, so…
I believe it comes down to showing that finite products are coproducts in additive categories, i.e. categories satisfying the first three of your axioms (in light of the first lemma here, for example).  This has been answered, for example, here:
In an additive category, why is finite products the same as finite coproducts?
