Examples of results which were surprising but later shown to be natural. 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? 
 A: In his Indiscrete Thoughts Gian-Carlo Rota writes:

Every mathematical theorem is
  eventually proved trivial. The
  mathematician's ideal of truth is
  triviality, and the community of
  mathematicians will not cease its
  beaver-like work on a newly discovered
  result until it has shown to
  everyone's satisfaction that all
  difficulties in the early proofs were
  spurious, and only an analytic
  triviality is to be found at the end
  of the road.

According to Rota what you ask for is the normal case. He - and others - do not even rule out the possibility that some day Fermat's Last Theorem turns out to be "trivial".
A: A favorite example of this kind for me is the result that the set of triangulations of an n gon (or, equivalenty, the set of interpretations of the nonassociative product $x_1x_2\dots x_{n-1}$) has the structure of a convex (n-1)-dimensional polytope. (It is called the associahedron or the Stasheff Polytope.) This looked (to me) as a curiosity at first but it turned out to be very basic and natural construction which is related to a lot of exciting mathematics. See Ziegler's lecture on the associahedron.
A: Gödel's incompleteess theorems -- knocked over the Hilbert program to prove mathematics was consistent and complete.
Smale's sphere eversion theorem - originally thought to be a counterexample showing an error in the proof, but the eversion is really there.
Butterfly effect (Edward Lorenz) -- looked like a numerical artifact in an ODE integrator, the idea of nonperiodic solutions hadn't been forseen.
Banach Tarski paradox
Monty Hall problem (ducks head)
Barrington's theorem for branching programs
PCP theorem
Rationality of Legendre's constant ;-)
