Expanded from an earlier answer of mine

The prevalence of the French language in Russian elite circles did not originate with the latter nineteenth-century Franco-Russian alliance, but had much deeper roots back to the eighteenth century. Particularly under Catherine II's direct patronage, French language and cultural forms eclipsed German as the main mark of European culturedness for much of the nobility and educated groups inside the Empire. Not only was it common for noble families to engage in converse and correspond in French, but many elite institutions such as Moscow University or St. Petersburg's Smol'nyi Institute for Noble Maidens conducted many of their classes, such as geography, in French. Learning French was a marks of status in elite circles, but also carried a significant political component to it as well.

One of the important things to realize of the Russian Empire's nobility was that a good portion of them were not ethnically Great Russian. Petrine imperial expansion not only brought in new elite groups to co-opt, like the Baltic Germans, but also attracted a number of foreign-born nobles into Romanov service. The linguistic composition of the empire was already multilingual and its nobility was no exception. While a working knowledge of Russian was needed, especially if one of these nobles entered into state service such as the army, knowledge of either German or French was necessary to navigate various social circles (this was also reflected in one of the truisms of Russian studies used to be an aspiring historian had better know Russian, German, and French or they would be lost). By the late-eighteenth century there were journals in French and German catering to the multilingual tastes of the few who could read. Russian generally took a second place to these other languages, much to the chagrin of various slavophile cultural critics.

But there was more than pragmatism that led to the prominence of French. Although Peter I's language policies had prioritized the knowledge of Dutch and German, starting in the reign of Elizabeth I, French started to eclipse other European languages as the language of culture and high society. Elizabeth's alliance with France certainly opened up the door to French culture and language, but it was arguably the Enlightenment that had a greater impact. Under Catherine II, French became very closely associated with ideas of progress and Europeanness, and court patronage of the philosophes gradually seeped down into other aspects of elite culture. While Enlightenment precepts of freedom could only go so far in a Russian imperial context, French theatre, novels, and poetry all penetrated the Russian elite cultural sphere and gave French a particular cachet. Catherine II embraced notions, albeit selectively, of the Enlightenment's call for reason and progress. Such a position on French was in keeping with the early Romanovs' larger project of both modernizing and Europeanizing Russia.

Of course, such Francophilia was not just limited to the Russian Empire. Throughout Europe in the eighteenth century, the French language and culture enjoyed enormous prestige. In Prussia, Frederick II extolled French as decidedly superior to German in art and grace. One of Voltaire's duties when he was employed by the Prussian King was to correct Frederick II's rather error-filled French writings, a duty the great philosophe refereed to among friends as washing the King's soiled laundry. Even across the Atlantic, a good many of the planter aristocracy in the South saw French as the language of polite society. Thomas Jefferson, for example, was sure to instruct the tutor for his daughter Patsy to have a good portion of her education in the French language.

The importance of French in Russian high society did cause some cultural critics to rise to the fore and defend Russian as a European language. Denis Fonvizin's comedies like The Brigadier-General often satirized the nobility's Francophile pretenses, even as Fonvizin borrowed French dramatic forms and some of his jokes required a knowledge of French. Other pedagogues and intellectuals began to articulate critiques of the frivolity of the Gallic-minded gentry to push for a renaissance of Russian letters and language. Even the Tsars gradually began to promote Russian in the nineteenth century and subtly critique the cosmopolitanism of the nobility. Tolstoy's War and Peace was reflective of the long-term push back against French and Francophilia as the novel contrasts two cultural Weltanschauungs: the technical and alienating French and the more down to earth and improvising Russian.

But French never really lost its position in Russian elite circles all the way up to 1917. Even after the invasion of Napoleon, few Russian elites were willing to toss out French in the name of patriotism. One of the professions French POWs had in Russia between their repatriation in 1814-16 was to act as French tutors to elite Russian families. French continued to be a major internal language in various state bureaucracies such as the foreign service, so French proficiency still had a major practical component to it. Additionally, the impact of French cultural mores was cumulative over time. A whole generation of Russian writers incorporated French ideas, motifs, and genres such as realism and then repackaged them as Russian. Even as Russian letters grew more Russian in content, there still was a strong French cultural component to them as there still was a deep association between France and the "West" in Russian high culture. The Alliance and the still glowing embers of cultural prestige attached to France ensured that the French language remained an important aspect of Russian elite education long after the Enlightenment.

Francophilia was a one-way street to an extent. Russian culture seldom occupied a place of prestige in French elite circles during the first half of the nineteenth century. The memory of 1812 in France often reduced the Russians into another component of France's defeat by nature, not as humans with agency and independent thought. Such sentiments that Russia was only a demi-oriental place had a long pedigree in France; Voltaire took his patronage by Catherine II as a sign of the force of the Enlightenment to transform even the most backwards of regions. The region of Eastern Europe that exercised the French imagination in the early nineteenth century was not Russia, but the disappeared Poland. In terms of music, the polonaise became something of a fad in this period and Polish emigres from Congress Poland like Frédéric Chopin found a home in France as did other figures of the Polish Romantic movement. Napoleon III used his uncle's patronage of the Duchy of Warsaw to tack to a more Polonophile cultural policy. Although enthusiasm for Polish culture in France died down by after Napoleon III's fall, it took some time for Russia to penetrate the French cultural sphere in place of Poland. By the early twentieth century, Russian arts became associated inside France with the avant-garde such as Stravinsky's The Rite of Spring or the various novelists of the Silver Age.

But French enthusiasm for Russian art was not quite Russophillia. Older demi-orientalist ideas about Russia remained in place as a half-civilized polity. Russia may have been France's ally against the Boche, but older tropes of a semi-civilized Russia grasping for hegemony over Europe remained quite strong in France as well as Britain. Notions of an ambitious Russian colossus arguably undergird the Alliance as it summoned up images of a Russian steamroller unthinkingly crushing Germany and Austria-Hungary.

One of the more remarkable features of the Franco-Russian cultural relationship over the long nineteenth century was that not much actually changed beyond subtle shifts. Francophilia remained a preoccupation of the elite almost to the point of satire while French letters tended to recapitulate older stereotypes of Russian inconsistency and un-Europeanness even when examples like Stravinsky proved otherwise. Even though the geopolitical relationship between the two countries had shifted dramatically by 1900, their cultural relationship had a marked tendency to be trapped in the eighteenth-century past.

Sources

McNally, Raymond T. "The Origins of Russophobia in France: 1812-1830." American Slavic and East European Review 17, no. 2 (1958): 173-189.

Offord, Derek;Ryazanova-Clarke, Lara, French and Russian in Imperial Russia. Edinburgh: Edinburgh University Press, 2015.

Wolff, Larry. Inventing Eastern Europe: The Map of Civilization on the Mind of the Enlightenment. Stanford, Calif: Stanford University Press, 1994.

Wortman, Richard S. Scenarios of Power: Myth and Ceremony in Russian Monarchy from Peter the Great to the Abdication of Nicholas II. Princeton: Princeton University Press, 2006.