I challenge the premise of your question. French authors indulged in a great deal of composition with Arthurian themes, but they were not necessarily the source of Arthurian tradition - including the legends told of Arthur and his court. In fact, Arthur was (and is) so intimately tied to Britain, that one could ask why French authors were attracted to this body of folklore, rather than why the English embraced the tradition.
Arthurian folklore has deep roots in Wales and Cornwall. If we accept the controversial conclusions of O.J. Padel, Geoffrey of Monmouth (c.1095–1155), author of The History of the Kings of Britain, came from Cornwall and Cornish folklore was also behind the Tristan cycle of Arthurian literature: Geoffrey of Monmouth is one of the first authors to write extensively about Arthur. We can add to this the late fourteenth-century "Sir Gawain and the Green Knight", which demonstrates that by the medieval period, England had also embraced Arthur and his court.
The question, then, is more about how the Arthurian tradition spread to French writers - and to other authors (for it was also popular in the Germanies and elsewhere). One can look to Brittany with its close ties and common language with Cornwall as a possible path for Arthurian tradition to reach France, but one must keep in mind that until fairly late, parts of France were held by the King of England. That resulted, of course, in considerable conflict - notably to the Hundred Years War (1337-1453), but all that was in response to those close ties, so it is not surprising to see shared traditions and literature.
Sources: O.J. Padel, ‘Geoffrey of Monmouth and Cornwall,’ Cambridge Medieval Celtic Studies, 8 (Winter 1984), pp. 1–28. O.J. Padel, ‘The Cornish background of the Tristan stories’, Cambridge Medieval Celtic Studies, 1 (Summer 1981), pp. 53–81.
edited to add date for "Sir Gawain"
Well, I'd been looking to get back in the game, and a medieval literature question right now is just too good to pass up even if it required a walk to the library in the dark and rain (thanks Scotland).
So the early textual history of Arthur is a mess, but I think it's a bit misleading to describe the Arthurian legend as "largely French." As some removed commentators noted, Arthur's genesis was in Brythonic folk tradition. What this means, is that in the 12th century when Geoffrey of Monmouth publishes his Historia Regum Britanniae, the first major Arthurian text, there are some major cultural crosscurrents that defy the simple English-French conflict. Firstly, Brythonic roots meant that in addition to a folk tradition on the British island, there was also an Arthurian tradition in Brittany, meaning that the story existed on the continent from the early days. Brittany was also directly adjacent to, and for some time ruled Normandy, which famously produced the french-speaking rulers of England after 1066. I know nothing close to enough to speculate about who would have known what about Arthur pre-Geoffrey, but it is useful to keep in mind that even from the early days, the national valence of the story was ambiguous.
That brings us to the famous chronicle of Geoffrey of Monmouth, the History of the Kings of Britain. Written in Latin, in Britian by a Cleric whose ethnic background is unknown, (Norman, Breton, and Welsh origin have all been floated as I understand it, and we'll probably never know.) the Chronicle was wildly popular. We have over 200 manuscripts containing it, which is a positively immense number (Sir Gawain and the Green Knight, the most famous English Arthurian Romance, survives in one). The History's particular popularity in England might have something to do with the fact that it had something for everyone. To the Welsh and Cornish, it told a story of their national glory, for the English, it told a tale of the rulers of the island being displaced, in time, by conquerors, and for the Normans, it grounded their crown in antiquity (Monmouth attributes the founding of the British kingdom to Brutus, the legendary son of Aeneas, the similarly legendary hero of Troy and founder of Rome) and helped legitimize their rule over the island. By the end of the story, it is clear that the story of the Britons is over as King Cadwallader is sent by an Angel to die in Rome, and the Saxons are clearly established as unjust and wicked conquerors. The Historia implies that the Normans are much like the textual Romans, conquerors of a barbarous people who will be sublimated into the superior culture, and furthermore the large number of Breton lords under William I provided another connection.
Monmouth's popular work was also adapted by an Anglo-Norman guy called Wace, whose Roman de Brut was a French "translation" of the Latin original. Wace substantially revised the work, expanding the Arthurian section, but what's important here is that Wace's adaptation and Geoffrey's original would be accessible to elite readers on both sides of the Channel. Norman elites still spoke, and if they could read, read French, and Latin was a transnational prestige language, and would remain so for centuries. I'll spend less time on him, other than to briefly mention that he originated the Round Table in the written works, though the sources I'm looking at suggest that it had folk tradition roots.
These and some other works make up the key dimensions of the "Chronicle" strand of the Arthurian tradition, and it would be these works that dominated the English Arthurian tradition, particularly in the earlier Middle Ages. The key texts of this tradition were clearly from the British Isles, (Terminology is hard here, at this time English can refer to either Normans or Saxons, and British can refer to all inhabitants of the British Isles or to Brythonic peoples") and they provide a sweeping history of Celtic Britain, and a dynastic hero equal to Charlemagne that the Plantagenet court could easily appropriate.
The other strand of the Arthurian tradition is what is known as the "Romance" strand, and this is the strand that is more closely associated with France, though it should be noted that the England-Chronicle/France-Romance duality shouldn't be overplayed. Wace and Geoffrey and other chronicles were read in France, and there are Arthurian romances in English. The French never had the same political interest in the "historical" Arthur, nor was Geoffrey's history taken quite as seriously in Paris as it might have been in England. However, the work was no doubt entertaining, and much of its material was adapted by Chretien de Troyes into his Arthurian romances. These and their french decedents share a number of features: A focus on Chivalry and the quests of Arthur's knights, rather than Arthur himself, a more ambiguous view of Arthur as a king (He usually isn't much of a player, and sometimes isn't an ideal king), and an increased focus on courtly love and the supernatural.
This tradition filtered back into England, with works like Sir Gawain and the Green Knight or or the various Death of Arthur books carrying substantial artistic debts to French Romance (and indeed, the modern Arthur of Camelot, with a round table of honorable knights questing for the grail, probably owes more to this Arthur than to the Chronicle one), but it was never really seen as undercutting the "Britishness" of Arthur. Nor did the French seem to have much trouble with Arthur either. While French audiences seem to have been a bit more skeptical of Geoffrey's historicity, they already had a perfectly good national hero, and so Arthur wasn't particularly threatening, and he provided an excellent source for lots of delightful romances!
This necessarily glosses over a LOT (I am not an Arthur scholar at all), particularly Arthur's reception outside of Britain and France, the continuing Brythonic Arthurian tradition, and some of the complexities of the genre interplay between Epic, History and Romance in Medieval Literature, (which is just too heady for me to cover tonight) but I think gets to the point of how Arthur could be popular in both Britain and France. For the former, he could be a historic, national hero, for the latter, he could be source material for tales of chivalric glory.
Sources:
Barron, William R. J., editor. The Arthur of the English: The Arthurian Legend in Medieval English Life and Literature. Rev. ed. with an additional postscript, Univ. of Wales Press, 2001.
Burgess, Glyn S., editor. The Arthur of the French: The Arthurian Legend in Medieval French and Occitan Literature. Univ. of Wales Press, 2009.
Ingham, Patricia Clare. Sovereign Fantasies: Arthurian Romance and the Making of Britain. University of Pennsylvania Press, 2001.
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Follow on question - given that the boundaries of England and France were somewhat in flux during this time, was the Arthur legend thought of as "ours" or "theirs" by English subjects on the (now) French mainland?
How did the Arthur legend, which has modern connotations of national identity, fit into the cultural identity of pre and early modern English subjects, especially commoners?