I'll try to keep it simple :

France (more precisely the territory of Gaul, so roughly actual borders of France) was before the first century BC inhabited by gauls, a disparate ensemble of continental celts who spoke the various dialects of the gaulish language, which was part of the continental celtic language continuum.

In the first century BC, the place was conquered by the roman empire, and integrated in the roman empire. Elites then the common people spoke at first a mix of both languages, then both variants of latin (vulgar latin and "classical" latin). The roman "occupation" lasted til the end of the roman empire proper, so roughly six centuries. That's a pretty long time, and by the end gaulish functionnaly disappeared.

In the beginning of the medieval era, Gaul was divided among 3 main germabic tribes, who spoke germanic languages : Franks who spoke Frankish, Burgundians who spoke Burgundian, Visigoths who spoke Gothic.

But these new germanic rulers were an infinitesimal part of the population (roughly 2-3% of the territory's population in the VIth century), who were heavily latinized in their ways of ruling, and had to intermarry with the local gallo-roman elites to survive and keep their power.

By the middle of the middle ages the local tongue of the common people was simply vulgar latin that evolved with some frankish/burgundian/gothic words but stayed fundamentally a latin-based language. As the elites weren't numerous enough and didn't had the will nor the number to impose their language, they took the local language instead.