TITLE: Crux of Dwork's proof of rationality of the zeta function?
QUESTION [24 upvotes]: As the question title suggests, what is the crux of Dwork's proof of the rationality of the zeta function? What is the intuition behind the proof, what are the key steps that the proof boils down to?

REPLY [7 votes]: Sasha has already pointed you to the primary source I used some years ago for my "expository" undergraduate thesis on Dwork's Theorem: Koblitz's p-adic Numbers, p-adic Analysis, and Zeta-Functions, which is available in 166 searchable pages here.
I've uploaded a copy of my thesis, which should constitute a relatively easy to digest ~ 50 page write-up of the proof. There is no significant difference from Koblitz's book; I worked out some of the pieces that were left as exercises or examples (e.g., showing that the zeta function for $f = x_1 x_4 - x_2 x_3 - 1$ is rational i.e. $\frac{1-qT}{1-q^3 T}$, which I re-produced in MO 117904) and omitted some of the proofs of cited theorems (e.g., the $p$-adic Weierstrass Preparation Theorem) with pointers to Koblitz's book or Gouvêa's intro to $p$-adic numbers when necessary.
As I understood matters almost a decade ago, there were a couple of important linear maps mentioned by Koblitz (included in my section 3.2) $G$ and $T_q$, which were composed to give the trace operator, $\Psi$, already mentioned. I introduce a slight bit of terminology ("admissible" in Definition 3.14) to discuss the trace of a linear operator from a certain infinite dimensional vector space to itself, and one of the key facts drawn upon in working towards the end of the proof is the power series identity:
$$
\det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right)
$$
for a linear operator $\psi$ going from a vector space to itself. I re-mention that identity here, because I asked about its history in MSE 277124 for which Markus Scheuer gave a great answer. This identity arises in proving that a sufficient-to-consider modification of the zeta function $Z'(T)$ is $p$-adic meromorphic, which is established through a bit of induction, analysis of the above identity, and the inclusion/exclusion principle (the last of which re-arises, as Sasha mentions, in proving Dwork's theorem for affine hypersurfaces can be extended to projective varieties).
I do not believe that there is anything novel in my undergrad write-up, but I think it is possible that its presentation as being intended for college juniors/seniors could make the proof, hence the underlying intuition, easier to grok.