The short answer is that they didn't. The Franks never were more than the elites of France, and the French are no more Franks than the English are French. The language of the Franks has a modern descendant in the form of Dutch, a rather Germanic language. (Do note that French does have a relatively high degree of Germanic influences compared to other Romance languages, due to the relatively large Frankish influences on early France, much like how English has a relatively large Romance vocabulary due to Norman influences on England.)

Furthermore, it's important to remember the borders of the Roman Empire, and the degree of control they had over the various regions. Look, for example, at this map of the Roman Empire at its greatest extent superimposed over modern borders.

What would end up becoming France was under Roman control since the days of Caesar, and remained under Roman Control for centuries later. Gallo-Romano culture had a significant presence in what would end up becoming France, and the Franconian Monarchs (like many rulers of the time) used the language of the Church (Latin) in formal documents, so there wasn't too much of an incentive for people to move away from Vulgar Latin (the spoken Latin of the time.)

Meanwhile most of the Germanic Tribes lived beyond Roman borders, and even the ones that lived in Roman territory had frequent contact with Germanic peoples beyond the border. Not to mention that once Roman power started to wane, the Germanic border regions were some of the first to be lost. There wasn't much room for a Germano-Roman culture to gain nearly as much traction as Gallo-Roman culture did in what would end up becoming France.

Finally, the concept of Holy Roman Empire wasn't really tied to Latin. Neither of the two states that claimed to be the successor of the Roman Empire (the HRE and the Eastern Roman AKA Byzantine Empire) spoke a Romance language in day to day life, and neither thought this was needed to validate this claim.

Edit: Removed questionable linguistic sidenote.

hi! fyi, you'll find lots more on the Franks' transition to French (but nothing on 'other Germanic tribes') in the FAQ

• Frankish and French

Confused - why is Vulgar Latin so far removed from Church Latin? It seems (to my uneducated ear) like the roots of French words are wholly unrelated to Latin. If I go to Italy, I may be able to make out some words based on my understanding of Spanish, but French seems a wholly unrelated mystery language. Can you explain how they became so different? Thanks!