What was Albert Einstein's proof for $E=mc^2$?

You can find the shortest and easiest derivation of this result in the paper where it was released by Einstein himself (what better reference can you find?) in 1905. It is not the main paper of Special Relativity, but a short document he added shortly afterwards.

A. Einstein,Ist die Trägheit eines Körpers von seinem Energieinhalt Abhängig?, Annalen der Physik 18 (1905) 639. A pdf file of the English translation Does the Inertia of a Body Depend upon its Energy-Content? is available here. (hattip: user53209.)

It is a delightful document to read. There is no dramatic references to huge power release nor anything similar. He simply states after the derivation "If a body gives the energy away $L$ in form of radiation, then its mass decreases in an amount $L/V^{2}$ (...) the mass of a body is a measure for its energy content (...) One can not exclude the possibility that, with the bodies whose energy content changes rapidy, for example radium salts, a proof of the theory will be found (...) If the theory adjusts to the facts, then the radiation transports inertia between emitters and absorbers."

Google for that short paper and see the derivation yourself, it is very easy. The Minkowsky four-dimensional spacetime had not yet been incorporated to special relativity, so the equations are formally very simple, easy to follow with little mathematical training.

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, Wikipedia has an excellent article on the history of relativity.