reading the classic papers, Khovanov-Lauda, Rouquier, Varagnolo-Vasserot, to dig out how to setup the convolution algebra (in general)

I don't even understand why these things can be realized using Fukaya category and Floer theory, besides 'physics motivation'. What's the math motivation?

Question 1: where does convolution algebra come from? Like Hecke algebra? What is the physical setup? you can say, some correspondence and push-pull.