After Ramanujan formulated his conjectures on the Tau-function, and after the importance of the function was realized, it took the development of the theory of Modular forms for the complete resolution and understanding of the conjectures and the function itself.(For example, it was only later that people could explain the appearance of the mysterious index 24 in its definition.)
Another example is the problem of constructibility of regular polygons. The Ancient Greeks must have pondered the reason for their inability to construct certain polygons. But after Gauss, it now seems natural why one cannot construct a 11-gon using only a compass and straightedge.
In the above two cases, there is a common feature. There is a discovery which at first seems surprising or baffling. Only later, after sufficient developments in theory, was the mystery lifted. Are there any other such examples?