Following the discussion at meta.MO, I'm going to post a good answer from the comments (made by JT) as a "community wiki" answer. I should mention that the Rogawski article mentioned by Tommaso says almost nothing about the proof of Ramanujan's conjecture, but it seems to be a very nice introduction to Jacquet-Langlands.
Deligne reduced Ramanujan's conjecture about the growth of tau to the Weil conjectures (in particular, the Riemann hypothesis) applied to a Kuga-Sato variety, in his paper Formes modulaires et representations l-adiques, Seminaire Bourbaki 355. I believe Jay Pottharst has made an English translation available.
Deligne then proved the Weil conjectures in his paper La conjecture de Weil. I.
As far as I know, all known proofs of this conjecture involve the use of cohomology of varieties over finite fields in an essential way.