Using Diamond, I have read and understand the definitions of modular curves as quotient groups of the upper half plane, modded out by congruence sub groups of $SL_2(Z)$, I understand that they can be interpreted as Riemann surfaces, then compactified, and as such, have an algebraic representation. The algebraic modular polynomial when equated to 0, has solutions that are the j-invariants, and so the solutions represent isomorphism classes of elliptic curves.
My question is this, I'm trying to present the above ideas and I did not really get a good understanding of why it is the case that we can go from working with quotient groups of the upper half plane, to parameterizing l-isogenous elliptic curves for example. I would greatly benefit from an explanation of the intuition behind all of the technical details and definitions.
Thanks!