HamtaroWarrior's claims that it doesn't have much to do with anime and that "It just seems to be a way to translate "Itadakimsu"." [sic] miss the point.
There is, in fact, a clear anime-related reason that "rub a dub dub, thanks for the grub" has become emblematic of over-localized translation: the abortive [gg] fansub of Toradora!! back in 2008 featured this very line as a translation of Minori's "itadakimasu" in episode 4 (around 08:52). This is an absurdly over-the-top translation choice, as any American could attest - Minori is certainly a cheerful and quirky character, but this goes well beyond that.
This was widely hailed as an innovation in trollsubbing on [gg]'s part, and was one of the factors that cemented their reputation as the leading trollsubbers of their time. I mentioned that the fansub was abortive - indeed, [gg] stopped subbing Toradora!! after episode 4, and I guess they decided to go out with a bang (all of the relevant blog posts I'd like to cite appear to have bitrotted out of existence, unfortunately). I emphasize this again - almost surely nobody actually thought that "rub a dub dub, thanks for the grub" was a good choice here. Rather, it was presumably chosen to get a rise out of its viewers (and succeeded quite well at it!).
It is as a result of this particular incident that images like this one exist (the "gg" column, at least). Likewise, the Sasami-san@Ganbaranai image in the question is a deliberate "homage" (if you can call it that) to [gg]'s Toradora!! subs of auld.