I cannot speak for all cases, but it was pretty common for most European nobles to learn a vast amount of languages compared to modern times. Almost every single man who was part of the social elite in Western Europe would during the early modern age be able to both read and write Latin. Especially monarchs typically knew quite a few languages if they had been raised to become a monarch, since they were expected to deal with the foreign affairs of the kingdom. Many noble families were also made up of mixed nationalities.

As an example the danish king Christian IV (1577-1648) spoke Latin, French, Danish, German and Italian. It wasn't even uncommon for many Danish monarchs to be better at German than Danish. King Christian would have been able to communicate with almost all other European monarchs at the time himself, at least in Western Europe, I don't know anything about Russia or other East European countries.

Addition: in many upper middle-class families in Britain and Russia in the late 19th century we see people who have studied and mastered English (in Russia), French, German, and some or a lot of Latin and/or Greek. This was simply expected of those strata. Many artists (including Turgenev, IIRC) lived in Paris for years, as many before and after did.

From the 17th century to 20th century (until WW1), the language of diplomacy was French. By Louis XIV's reign, French had replaced Latin as the language for diplomats but also for cultural/artistic and intellectual productions. This was due to the influence France had in Europe, partly because of the important literary production (Molière, La Fontaine, Bossuet). Non-French writers sometimes published their work in French and not in their own language and intellectual society wrote and spoke in French. For example, the periodical literary review Nouvelles de la république des lettres was written in French for a European audience.

European royals knew in most cases the main language of their country (I say "main" because regional languages were common, especially in France for example), they were also taught Latin for religious purposes. Greek was rarer because it targeted a more intellectual audience and sometimes another language. Louis XIV could speak Italian (which dialect I do not know) quite well. George I of England, born in Hanover had a hard time bothering to learn English. Louis XVI could speak, read and write in English very well (he read the English press).

The 18th century was the heyday of the French language in Europe. It was spoken at Frederick the Great of Prussia's court. Catherine the Great of Russia spoke her native German language but everything in the Russian court was in French. Some Russian aristocrats did not know Russian (increasing the gap between them and their people) at all. As an example, when diplomat James Harris (future Earl of Malmesbury) was sent to Frederick the Great and later Catherine's courts, he spoke in French and not in English, German or Russian.

There are some exceptions, of course, most of the written diplomatic relation between France and the Holy Roman Empire remained written in Latin as a tradition but oral communication was in French. Ambassadors and intellectuals communicated in French.

In 1248 the French king Louis IX used interpreters during his crusade, according to French chroniclers. While he was in Egypt, Bandouin (Baudoin) d'Ibelin interpreted the words of the Saracens, and another interpreter, Nicole d'Acre, is also mentioned in the same source (chronicles of Joinville). The use of interpreters in royal businesses goes back at least to the thirteenth century.

While Louis IX was able to find people who knew Arabic and could assist with interpreting in the crusader kingdoms, in other cases slaves taken originally from one country to another at a young age could be used as bilingual interpreters.

Jean Delisle, Judith Woodsworth, eds. Translators through history, John Benjamins, 2012, has a chapter on "Interpreters and the making of history" that may be of interest.