This is a result of Lickorish, in his paper "A representation of orientable combinatorial 3-manifolds".  The paper is only eleven pages, and is very readable.  One important remark: along the way Lickorish rediscovers some ideas first investigated by Max Dehn.  This result was definitely unexpected at the time; I don't think that Witten's remark is really justified...

Here is a very sketchy overview of Lickorish's paper. 

The first step is to note that every closed, connected, orientable three-manifold $M$ has a Heegaard splitting.  That is, you can get $M$ by gluing together a pair of handlebodies.  (That this is true for combinatorial manifolds is an easy exercise.  That all three-manifolds are combinatorial is a very difficult theorem.)

The second step is to prove that any orientation-preserving homeomorphism of a closed surface is isotopic to a product of Dehn twists.  This requires proving that you can get from any (non-separating) curve to any other by a chain of curves, with each pair meeting exactly once. 

The third step is to relate Dehn twists to a special case of Dehn surgeries - this is probably the most "three-dimensional" part of the proof. 

Since the three-sphere has a Heegaard splitting in every genus, the theorem will follow.