I find it confusing mostly because the language name French is clearly based upon the people who used it the Franks, but they were Germanic, so surely they would have originally spoke a Germanic language.
The Franks were a group from what is vaguely now Germany which took control of Roman Gaul in the fifth century. However, when they took control, they were vastly outnumbered by the people who were already there, who, of course, all spoke Latin. The Franks did not kick these people out (which would have been an impossible undertaking), nor did they really overthrow the existing government, preferring rather to co-opt the Roman administrative bureaucracy. Thus, the majority of the people in Gaul still spoke Latin, which eventually evolved into French.
Latin was actually quite widespread, extending beyond the firm boundaries of the Empire. Even in regions which are today Germany, Switzerland, and Austria, which generally spoke a Germanic languages, have attested islands of Romance speakers into the 9th century, a fact which still remains attested in some local dialects today.
The definitive texts on the subject are, unfortunately, in French:
Norberg, Dag Ludvig. Manuel pratique de latin médiéval. Paris: A. et J. Picard, 1968.
Riché, Pierre. Education et culture dans l’Occident barbare, VIe-VIIIe siècle. 4e éd., rev. et corr. Points H195. Paris: Editions du Seuil, 1995.
As always u/tekjanuru does an excellent job.
I will throw in a source, in English, that is worth reading.
D. H. Green's Language and History in the Early Germanic World Cambridge: Cambridge University Press, 1998.
It's focus is a bit more on writing and literature than on spoken language but its focus on the interaction between Germanic, Christian, and "Latin" traditions may be of some interest to you.
hi! FYI, there was a good discussion on the Franks recently,
How did the people of Gaul come to be known as Franks?
... and we've got a FAQ* section on how they ended up speaking French
*see the link on the sidebar or the wiki tab