What is the graphic intuition behind Marginal rate of subsitution
This isn't a perfect answer to your question, but I tend to show all my students pages 1-4 of Dixit's Optimization book. It describes exactly what the MRS is doing and why it must be tangent to the budget constraint. 

https://www.scribd.com/document/390573490/Dixit-Optimization-in-Economic-Theory-pdf (pages 1-3 here unless you sign in)