Following the discussion at [meta.MO][1], 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][2], 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][3].

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.

  [1]: http://mathoverflow.tqft.net/discussion/328/should-we-do-anything-if-a-question-is-answered-well-in-the-comments/
  [2]: http://www.google.com/search?q=formes+modulaires+et+representations+l-adiques
  [3]: http://www.numdam.org/item?id=PMIHES_1974__43__273_0

Added by Emerton: One point to make is that the Weil conjectures (in their basic form,
saying that the eigenvalues of Frobenius on the $i$th etale cohomology of a variety over
$\mathbb F_q$ have absolute value $q^{i/2}$) apply only to smooth proper varieties.  On the other hand, the Kuga-Sato variety is the symmeteric power of the universal elliptic curve over a modular curve, which is not projective.  Thus one has to pass to a smooth compactification in order to apply the Weil conjectures, and then hope that this does not mess anything up in the rest of the argument.  A certain amount of Deligne's effort in
his Bourbaki seminar is devoted to dealing with this issue.  If you don't worry about this
(i.e. you accept that it all works out okay) then the proof is essentially just Eichler--Shimura theory (i.e. the relation between modular forms and cohomology of modular curves), but done with etale cohomology, combined with the Eichler--Shimura congruence relation that connects the $p$th Hecke operator to Frobenius mod $p$. (The latter was treated in the
[following question](https://mathoverflow.net/questions/19390).)