This is a reference to the folk knowledge that you should only eat oysters in months that have an "R" in the name — based on the coincidence that September through April, the winter months, have "R" and May through August don't.
According to this article, possible reasons include the oysters' need to repopulate and/or their toughness during the summer spawning season, the presence of algae-related toxins during the summer, and the tendency of shellfish to spoil in the heat in the age before refrigeration, but it notes that this is not a concern with modern commercial oyster farms... I don't know; I'm not a cook. :)
Overkill, vol. 1: Do we owe the mnemonic to the French?
The discussion about whether this is a "French" question made me curious to find out whether this is a French phenomenon in particular. The Wikipedia article on oysters evenhandedly mentions:
It was once assumed that oysters were only safe to eat in months with the letter 'r' in their English and French names.
I soon found this article on the history of the belief, which mentions two sources and a fuller explanation thereof. The first source is the (possibly comedic) 1599 cookbook by Englishman Henry Buttes, Dyets Dry Dinner. Sure enough, in the "Oyster" chapter, we find this:
In those months that have the letter R. in their names.
The second source is a Latin maxim the article says was used in medieval times and is cited in an 1890 book. Searching for that maxim, I pushed the reference back further to a translation of Pliny the Elder's Natural History, cf. the end of this page and the top of the next:
... it is at the beginning of summer, more particularly, and when the rays of the sun penetrate the shallow waters, that they are swollen with an abundance of milk.
(The article corroborates this description by qualifying the summer months as spawning months, during which the oyster "fertilises its eggs inside the shell and incubates them about for 10 days before releasing them into the water [...] During this process its flesh often becomes quite milky".)
A translator's or editor's note on the Pliny passage alludes to the aforementioned Latin maxim:
We have a saying that the oyster should never be eaten in the months without an r; that the same, too, was the opinion in the middle ages is proved by the Leonine line: "Mensibus erratis vos ostrea manducatis." "In the r'd months you may your oysters eat."
That link doesn't give much information on the provenance of the translation, and so I wonder whether this "we" does not in fact refer to the English. Perhaps the note itself is translated from an original language. Consider that a similar passage occurs in an inline note in this 1829 book published in Latin in Paris (p. 493, halfway down left column).
Of course, Pliny doesn't actually mention the "R" rule, even though the distribution of "R" in month names is the same in English, French, and Latin. Besides, I see that Leonine verse was an invention of the European Middle Ages "foreign to Classical Latin poetry", grounding that particular formulation indeed long after ancient times. As the use of Latin was widespread in Europe during that time, it's hard to know whether that maxim would have been French, but that article just cited mentions implies that Leonine verse is tied to France being that two people associated with the form, Alan Rufus and Léonin, were both French.
It seems pretty likely, then, that Buttes' English cookbook was far from the originator of the idea that oysters must not be eaten in the summer. That was an ancient idea. Neither was it the likely originator of the "R" mnemonic itself. That mnemonic may well have been a French original or at least found early expression by the medieval French, speaking Latin.
That article on the history of the idea also associates it specifically with France (it's not a French website, so the mention is not inevitable):
the French have traditionally eaten a huge proportion of their oysters around December, at the height of winter and at Christmas
Finally, we can also conclude pretty definitively that in this New York Times piece, which set me on the wrong track for a while, the word "introduced" at very least is nonsense:
American Indians are said to have introduced this warning centuries ago to early settlers...
Overkill, vol. 2: How do we interpret the French?
Finally, to make this a truly French question from one more angle: how do we interpret the actual phrase un mois en "R" ? Of course, we know what it means by now, but what if we didn't?
To my mind, what exactly would qualify a month as being en "R" is at best implicit. I suppose there are few other possible connections between the letter "R" and any given month besides its name, but the facility with which that step might be made didn't stop our 1829 authors from feeling the need to clarify what was presumably a neologism in the Latin maxim:
Mensibus erratis (quorum nomini inest littera r) vos ostrea manducatis.
In R'd months (in whose name the letter r is present) may you eat your oysters.
(Is un mois en "R" any clearer than, say, un mois rhotique would be if that parallel to the Latin adjective were used here?)
What exactly does en communicate here? It evidently doesn't have the sense of "in", nor "into", "while", "during", "sous forme de", nor even "dont le matériel est". It seems to be rather best expressed as "having", or better, "qualified by".
It's hard to find either of these senses in the TLFi entry. Entry I. A. 3. near the top is not that far off from "having" semantically:
Riche en blé. Pauvre en ressources.
Rich in (having) wheat. Poor in (having) resources.
But syntactically it's not close enough to our mois en "R".
If I were asked to choose a preposition to express "qualified by", I would rather choose à, which figures in many semantically parallel phrases, as in section III. D. 2. of its gargantuan TLFi entry:
armoire à glace
bateau à voile
boîte à casiers
bombe à ailettes
instruments à cordes
lampe à arc
However, Laurent observes below that one jokes about les jours en "di", and Feelow observes that one also says rimes en [sound] with the sense of "rhymes containing or consisting of [sound]". The latter becomes more compelling when we consider that, despite our 1829 authors' gloss, erratis might refer not to the presence of "R" in the spelling but plausibly to the sound "R" in the pronunciation.
Could this then be a special use of en?
Prepositions are difficult to pin down semantically, and a full inquiry would be out of the present scope, but we could note three hypotheses that are not mutually exclusive:
Sometimes a particular preposition collocates with a particular class of word. For example, English often matches "with" and an emotion noun to create a complement of manner: "with gusto", "with enthusiasm". Perhaps "en" collocates with sounds to mean "containing [sound]".
It's possible that "en" carrying this sense that seems more proper to "à" was more normal in the past but is not productive today.
Phonologically, en could obtain to facilitate liaison and avoid the quite difficult hiatus that would obtain in un mois à "R".
Since this last section is one for which a native speaker might have a better intuition, I invite commentary on it.