Putting math in historical perspective helps a lot. Like most students I was taught the Calculus with the rigorous epsilon delta definition of a limit which was then used to define derivatives. It never really clicked until I read the historical context for Newton. He didn't use limits, but used infinitesimals, which are less rigorous but make a lot more sense to me at least.
I totally agree -- epsilons and deltas were invented to deal with issues of rigor that were raised 100 years after Newton. In the meanwhile, he was getting stuff done with his "nonrigorous" Calculus.