French was the language of European aristocracy, nobility and royalty for most of (early) modern history. The Romanovs were, like all the major European houses, deeply interconnected and intermarried with the European nobility. This is why French was also spoken at the Russian court, because it was the language of the nobility.

Nicholas II and his cousins Wilhelm II of Germany and George V of the United Kingdom are great examples of how interwoven European nobility was. They corresponded and visited each other and their other relatives regularly. You can imagine that this necessitates a common language.

Emperor Nicholas II did speak English though, his famous pre-WWI correspondence with German Emperor Wilhelm II was conducted in English.