Einstein's equation doesn't have a "proof" because it's not a mathematical theorem.  It's a physical *theory* that is overwhelmingly supported by experimental data.  So you could say that the "proof" is in the mountains of experimental results that agree with the theory.

To understand Einstein's motivation for developing the theory of relativity, as well as [mass-energy equivalence][1], Wikipedia has an excellent article on the [history of relativity][2].


  [1]: http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence#History
  [2]: http://en.wikipedia.org/wiki/History_of_special_relativity