I've been thinking about your question for a couple of days, and figured that you can find a partial answer in the work of the historian of science Michael D. Gordin, specifically in his book the Scientific Babel. Why partial? Because Gordin focuses on a period (from 1880 to today) when the monotonous decline of French as a scientific language had already started. However, he gives a very informative answer as to how English had become so dominant during the 20th century.

Gordin's argument is essentially that up to the beginning of the 20th century, scientific communication was dominated by a triumvirate of European languages: German, English and French. He claims that the first impulse which shook the dominance of that triumvirate came from "the rise of nationalist ambitions from large scientific cohorts such as the Russophone one, which challenged the tight strictures around three dominant languages" (p. 306). However, this was just the initial push. The biggest reason behind the dominance of English, according to Gordin, was the combination of American geopolitical and financial power and their refusal, as the largest and richest Anglophone population, to continue learning foreign languages.

Another precipitating factor is the rise of computerized referencing tools (and the prestige economy which latched onto them). Gordin: "Database followed database, and the advent of hyperinfluential metrics such as the Science Citation Index and 'impact factors' only increased the first-mover advantage that had accrued to American indices. Publishing in English placed the lowest barriers toward making one's work "detectable" to researchers" (p. 308-309).

When reading about it, just the sheer size of scientific English, by the end of the 20th century, is staggering. At the very beginning of the book, Gordin discusses a graph from this article (unfortunately paywalled). The graph shows plotted percentages of the global scientific literature and the languages it's published in (German, English, French, Japanese, Russian). Gordin accompanies it with this comment (p. 6-7), commenting on how English became dominant in 1930, but seemingly plateaued in the dominance until the 1970s:

The situation is actually even more dramatic than it appears from this graph, for these are percentages of scientific publication - slices of a pie, if you will - and that pie is not static. On the contrary, scientific publication exploded across this period, which means that even in the period from 1940 to 1970 when English seems mostly flat, it is actually a constant percentage of an exponentially growing baseline. By the 1990s, we witness a significant ramp-up on top of an increasingly massive foundation: waves on top of deluges on top of tsunamis of scientific English.

All in all, that's the story of how English became dominant. Hopefully someone will come by and tell us more about the decline of French specifically.