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A fairly detailed (14 page) account of Apéry’s original proof of the irrationality of $\zeta(3)$ is given in Julian Havil’s book The Irrationals which states that Apéry’s starting point is the mysterious recurrence relation $$n^3u_n + (n-1)^3u_{n-2} = (34n^3 - 51n^2 + 27n - 5)u_{n-1}.$$ From this, he defines two sequences using two pairs of boundary conditions. He then gives, without proof, closed-form expressions for these sequences on which Havil comments “... and it becomes easy to appreciate the incredulity of those who attended that conference.” Finally, he proves that the term-wise ratio of these two sequences converges to $\zeta(3)$ and then uses Dirichlet’s irrationality criterion to prove irrationality.

Q: Did Apéry ever explain how he obtained this recurrence?

I have been able to locate two different derivations of this recurrence relation, however both appear to be more recent than Apéry’s work and indeed appear to be motivated by it. For example, in the article Irrationality Proofs for Zeta Values and Dinner Parties (page 15/32), regarding the Apéry recurrence relation it states “There are numerous interpretations of this recurrence relation as a Picard-Fuchs equation of a family of varieties.” The other derivation is via the relatively new concept of Cellular Integrals and Beuker Integrals - see, for example Elliptical Integerals, Elliptic Functions and Modular Forms in Quantum Field Theory, starting on page 342. (Beuker presented a simplified version of Apéry’s proof using "Beuker integrals".)

(nb: The Wikipedia page on Apéry’s Theorem makes no mention of this recurrence relation in its summary of the proof. However if you read Apéry’s summary of his proof (linked on wiki), then we see the two sequences he uses do indeed come from the recurrence.)

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    $\begingroup$ I probably should have mentioned in my summary of Apéry's argument that he introduces an auxiliary variable into one of the closed-form solutions, but for simplicity I omitted it. Also, upon reflection, I imagine that if Apéry had explained the origin of the recurrence relation, then Havil, who devotes a whole chapter to the result and 14 pages to the proof, probably would have mentioned it. $\endgroup$ – Nick Mar 28 at 15:18

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