Simply put, the Franks were a West Germanic people originating near the Lower Rhine, an area approximately in the South of the modern day Netherlands. They had so much time to adopt aspects of Roman identity that they adopted Christianity and their native Germanic tongue evolved into French. It was so many centuries that they were able to expand and contract in power so that there were cultural splits. Franks in Gaul slowly adopted the a more "French" culture, while Franks that had conquered and settled in Germany adopted a more "German" one. Frankish history is a favorite of mine, so I'll give a larger synopsis but I may miss some of the finer details on Ethnogenesis (how an ethnicity if formed). Over the centuries the lands they lived in and ruled had many names as linguistics and demographics evolved.

To start, the Franks were actually a collection of tribes (Saliand and Ripuarians being most well known) with various names, attested by the Romans in the 3rd century (200s AD) in Germania Inferior. They were living in proximity, and possible connection with the Ubii and Batavi. The Ubii and Batavi were rather amicable with Rome, many of whom served in the legions. However, as time went on the Franks would have both belligerent and amicable agreements with Rome, and their other West Germanic allies. At different times, some Frankish tribes would be friends of Rome, while others may be hostile. One must understand that along the Roman frontier there was a lot of both cooperation and low intensity hostilities until the later 4th and early 5th century (the 370sAD to nearly 500AD saw many migrating peoples moving into Roman territory).

After displacing some of the previous inhabitants, such as the Batavi, the Franks began moving closer and closer to Roman territory. By 358AD, some Franks were allowed to ally and settle in Roman territory West of the Rhine, known as Toxandria. They would then serve as "foederati", foreign raised tribal warriors in service of Rome. The Franks in particular would remain pagan for another couple hundred years, despite Christianity growing. Additionally, as time went on the power of the Frankish tribes had been consolidating, growing, and increasingly becoming Romanised. In the next hundred years of instability, the Franks would go on to occupy more and more Roman territory in modern day Belgium, Germany, and Northern France (Gaul). They were settling in what was still nominally the Roman Empire, but was under control of the local Frankish tribes.

In 451AD, Frankish warriors participated on both sides of the Battle of Chalons between Rome and its allies against Attila and his client peoples. Further propelling themselves into positions of regional power. In the coming years as migrating and conquering Germanic peoples deteriorated imperial Roman authority, the Franks descended upon all of Northern Gaul, including the Roman rump state of Syagrius. After Western Roman imperial authority collapsed, the Franks, now known as the Merovingian Franks began to convert to Latin Christianity at a much faster rate in the early 6th century AD (~511AD under King Clovis I).

Although Western Roman imperial authority had disappeared, many Roman people still existed in the Frankish Kingdom. They both praised and criticized Frankish rule, as on one hand they were definitely not Roman but they also weren't completely foreign as they attempted to emulate aspects of Roman authority. Over the next few hundred years, which would include the main expansion (Charlemagne's Carolingian Renaissance) and collapse of the Frankish Empire (Charlemagne's descendants successor conflicts). The Franks would consolidate authority to go from being a ruling people with a dominion over Romans and other Germans, to a ruling establishment. They expanded into nearly all of France (Aquitaine, Burgundy etc), Western Germany, and Northern Italy. The Franks that settled in France (West Francia), ruling over Latin Romans adopted French and Occitan as languages. The ones that settled in modern day Germany (East Francia) to rule over those that never had been Roman before continued to use Germanic languages. By about 900AD, the split was significant enough to say that these lands and peoples had different destinies. The Frankish Empire, built upon some foundation of the collapsed Roman Empire, established the foundations of government and identity for much of feudal Europe. As it passed into memory, the fragmented continuity of power developed into the Kingdom of France, the German states forming the Holy Roman Empire (which I didn't mention at all), and the Kingdom of Italy (which was also part of the HRE).

So there as you see, over the course of about 1000 years a lot will happen. The Franks began Germanic, but over time went on to develop into French (Latin + Frankish = French) and German (an evolution of old Germanic languages). It happened so long ago, over a large area there was a divides due to distance and regional identities that went on for generations.

The Franks: From their Origin as a Confederacy to the Establishment of the Kingdom of France and the German Empire by Lewis Sergeant

A History of Medieval Europe: From Constantine to Saint Louis by R.H.C. Davis

The Inheritance of Rome: Illuminating the Dark Ages 400-1000 by Chris Wickham

Two Lives of Charlemagne by David Ganz (written from more accounts contemporary with Charlemagne)

There's actually a lot more material on the history of the Franks too, and their laws were written down so there has been much study on it!

This question is a little complicated. The Franks were Germanic in origin (the Frankish language is a forerunner of modern-day Dutch) but did end up "Latinifying" after their conquest of Gaul. In the ninth century there are a couple documents referring to the language of the "common people" being either Germanic or "Romanz" (the Late Latin forerunner to the rest of the Romance languages), the line of division would have been roughly the old border of the Roman Empire. It's hard to tell exactly when Frankish fell out of use for Romanz in the western empire (modern-day France) since all the documentation is in Classical Latin. Eventually, however, the Franks did assimilate into their Romanz-speaking populace, though modern-day French still has a lot of Germanic influences due to the period of Frankish rule.