Renormalization is not the same thing as regularization. Regularization is a mathematical trick while renormalization has a very physical interpretation, for example, its use in statistical mechanics. In QFT, they are related and I think sometimes conflated, which is unfortunate.

You can easily describe renormalization without math because it's a physical concept. Regularization is more difficult. If the OP decided divergent integral regularization was an interesting conversation topic then I am on your side.