The math behind beginning of Quantum Mechanics is not hard. That's actually a lie, it is hard, we just make alot of assumptions and stuff to make it easier. Because of this however, I am sad to say that some mechanisms here can only be explained by "Just trust me bro". Much of these mechanisms are governed by Quantum Field Theory, where the math is unbelievably hard. Oh, Whatever you do, do not ask about relativity in Quantum Mechanics. I repeat DO NOT. Unless your body is ready for the horrors.
For Quantum Mechanics, the math is Calculus and Linear Algebra. If you got a 5 on the AP Calc BC exam, thats all the calc you'll need. As for Linear Algebra, you'll need to take a class. Or you can get all the knowledge for free online. This guide is for getting the math knowledge free online.
The science behind Quantum Mechanics started because basically physicists basically did everything else so they were like ok guys let's just wrap up the loose ends and pack it up. I still wonder what physicists were gonna do if they hadn't discovered Quantum Mechanics. What. go home? i don't know.