_{ Here is a relevant reference not mentioned so far, which in a sense (taking the strong statement I cite below at face value) gives an answer to the OP, and is so new that perhaps this is unknown to the OP. I am posting this since I just stumbled about this thread while reading the chapter that I quote from below. It definitely seems on-topic for this thread. }

In the very recent Sammelband on Gerhard Gentzen's legacy

Gentzen’s Centenary: *The Quest for Consistency*.
edited by Reinhard Kahle and Michael Rathjen
Springer, x+561 pages, Springer, 2015,
ISBN: 978-3-3191-0103-3

on pp. 5-6 of the contribution

[Reinhard Kahle: Gentzen’s Consistency Proof in Context. op. cit. p. 3-24]

one reads

"But Hilbert was, by no means, a philosophical hardliner. **The only piece of written evidence which we have about Hilbert's reception of Gödel's result** [emphasis added] is the cryptic short preface in the first volume of the *Grundlagen der Mathematik* [52], saying that Gödel's result "shows only that---for more advanced consistency proofs---the finitistic standpoint has to be exploited in a manner that is sharper [...],"^{11} i.e., the philosophical starting point was to change. **Bernays and Ackermann provide us with two additional testimonies that Hilbert soon adapted his "meta-mathematical standpoint."** [emphasis added] Based on Bernay's reports, Reid [whose writing about Hilbert *seems* to be thought of as substantiated by actual historical research; see e.g. the AMS obituary] writes about Hilbert's reaction to Gödel's result [ Constance Reid: *Hilbert*, p. 198] "At first he was only angry and frustrated, but then he began to try to deal constructively^{1} with the problem. Bernays found himself impressed that even now, at the very end of his career, Hilbert was able to make great changes in his program." Ackermann writes in a letter to Hilbert (August 23rd, 1933)^{12}: "I was particularly interested in the new meta-mathematical standpoint which you now adopt and which was provoked by Gödel's work." Unfortunately, we have no sources which explicate in detail Hilbert's new standpoint, but it goes without saying that Gentzen's work was in line with it.

I here give the footnote '^{11}' from the citation above:

"[Hilbert and Bernays, p. VII]. German original: "Jenes Ergebnis zeigt in der Tat auch nur, daß man für die weitergehenden Widerspruchsfreiheitsbeweise den finiten Standpunkt in einer schärferen Weise ausnutzen muß, [...]."

I here give the footnote '^{12}' from the citation above

German original [Kahle here gives a reference to a letter of Ackermann to Hilbert, apparently kept by the Universitätsbibliothek Göttingen]: "Besonders interessiert hat mich der neue meta-mathematische Standpunkt, den Sie jetzt einnehmen und der durch die Gödelsche Arbeit veranlaßt worden ist." The letter was written after Ackermann visited Göttingen, but didn't meet Hilbert and spoke only with Arnold Schmidt, who informed him about "everything" [Kahle does not indicate what sort of quotation marks these are, i.e., whether these are scare quotes meant to indicate that 'everything' is used ironically, or quotation marks indicating that someone is cited here] going on in Göttingen.

So, to summarize

(0) Kahle makes a strong statement 'the only written evidence we have' about what is, in a sense, only a semi-decidable problem (to think, e.g., of Hilbert's boxes Reid found in Göttingen), and this statement of Kahle's essentially says that what the OP quoted is the only historical evidence of how Hilbert reacted.

(1) The answer to the question "What did Hilbert mean by that statement?" in the OP seems to be: "No one really knows, yet much can be extrapolated from what Hilbert did in the ten years after this statement."

^{1} _{Reid's use of 'to deal constructively with the problem' is a little awkward in this context, for obvious reasons. Here, scare-quotes would have been in order.}