Well, IDK if I'll be able to explain it better than your teacher did, but it is essentially a way to make the integration process faster by replacing a part of the function with u, differentiating this part and subsequently writing both dx and the function in terms of u (and du). You just make it look simpler in order to understand what integration rule is applicable. Say, if you have (2x+3)^3, then you let 2x+3 be u; hence du/dx is 2; dx=du/2; 1/2 int(u^3du)= you integrate it in the regular way and get shit done. It can be more complicated with more complex functions, but that's the basic mechanism.