Prior to Peter I and Catherine II, Russian court language was a kind of Church Slavonic, and showed a mix of East Slavic features that we find in modern Russian along with a large proportion of South Slavic loanwords. To illustrate, a noble at court might have remarked that the weather was xladnaja 'cold', using a form you'd find in Church Slavonic, which was fundamentally South Slavic, (cf Slovene hladen), while a peasant in the countryside would have said instead that it was xolodnaja, using an East Slavic form.

So, that left Russian nobles in a sort of double-bind: on the one hand, they had their court language, which was heavily associated with the Orthodox Church--not exactly an institution Russian monarchs liked during the Enlightenment--and thus associated with 'backwardness'; on the other hand, there was vernacular Russian, which just didn't have a vocabulary that would allow it to be easily used in a court setting, and was also associated with 'backwardness', 'roughness', and other negative stereotypes of peasants or serfs.

French's place in the Russian court grew out of modernizing efforts begun by Peter the Great. He and later monarchs encouraged Russian nobles to learn foreign languages, and French was the lingua franca of diplomacy at the time and thus de rigueur for anyone at court. French had a lot of social prestige and cachet for other reasons as well: France and French speakers were a center of the Enlightenment, something with which both Peter I and Catherine II were pretty pre-occupied with. France produced a lot of the educational texts used in Russia, and Russian itself was not thought to be capable of handling diplomatic, scientific, or philosophical transactions--it was viewed as vulgar and coarse.

To be perfectly clear, this was merely a perception, and was not in any sense true--the language of peasants and workers wasn't used in court, obviously, and so lacked words that you might need or want in court, but intellectuals and writers like Lomonosov, Karamzin, and Pushkin did a lot of work in the late 18th and early 19th centuries synthesizing Church Slavic, vernacular Russian, and French, German, Greek, Polish, and Neo-Latin borrowings into what we might call Early Modern Russian.