I'm not sure there can be one single answer for this as it will differ from person to person, and it depends on the language and the area of history being studied. So hopefully this can be kind of a meta-thread and I can give some anecdotal evidence...

Basically the answer is...practise, practise, practise. You might need to be conversationally fluent in a modern spoken language or you might just need to be able to read documents. The circumstances will affect how you approach learning the language.

For my degree in medieval history, the requirements were that everyone (no matter what specific thing were were actually studying) had to pass written exams in French, German, and Latin. We would get 3 or 4 paragraphs about something relating to medieval history, and we had to translate them with a certain degree of accuracy (subject to the whims of the exam committee, really - the criteria for what was accurate or not were always secret).

French was easy - I'm Canadian so we learn French in school from a young age. The vast majority of people stop taking French as soon as they can and then they forget everything they learned, but I was one of the rare weirdo kids who kept taking French classes all the way through university.

Coincidentally I had also studied German and Latin on my own (again, weirdo kid) and then formally as an undergrad. The German and French exams were necessary because most medieval history books are in those languages (and English).

The Latin exam was actually two levels ("intermediate" and "advanced" but we always called them the MA and PhD exams), because so much literature written in the Middle Ages is in Latin, and we had to have to really good grasp of it. Almost nobody passed either level of the exam the first time, and I think we had 3 or 4 years to pass them, and there were two opportunities per year...something like that. Meanwhile there were Latin classes taught by the professors and the upper-year students who had already passed, and when I say the classes were intense, I mean INTENSE. We often described as being beaten over the head with Latin. You took the classes, read and translated vast amounts of Latin texts, and you kept doing it until you passed the exam. If you didn't pass the exam after the 3 or 4 years or whatever it was, you wouldn't be allowed to continue. Several people who started with me got kicked out after failing the exam too many times.

For some students it was a bit unfair - for me, French, German, and Latin are the perfect languages to study the crusades, but what if you study Middle English, or Old Irish, or medieval Russian? Sometimes Latin just isn't very useful for certain areas of medieval studies. But those were the rules, you had to pass these exams regardless of how useful they were for you.

Speaking of those other languages, once we had passed the exams, we were expected to take any other language classes that might be useful in our future careers, although they weren't mandatory. I took Old French and Arabic and they were the same idea - intensive reading and translating medieval texts. Other students took Italian, or Greek, or Hebrew, or Old Norse - there were plenty of options available (I wish I had taken Greek!). Of course I doubt most people learned every language equally effectively - personally my German and Arabic are pretty bad, they suffered from focusing more on Latin and modern/medieval French.

That isn't unexpected though, nobody can expect you to become perfectly fluent in every language you have to study, especially if there are so many of them. And not every graduate school program is like this; mine apparently has a terrifying reputation (especially for Latin).

Over the in the classics and Near/Middle Eastern studies departments there were even more dead languages to study. I have a friend who studied Egyptian, Akkadian, and Sumerian. No one expects you to learn to speak Latin fluently (although some people do), and there's certainly no way to speak conversational Sumerian!

Fortunately for native English speakers, it's not hard to get by in the academic world only speaking English. Lots of academics speak English as a second language. As long as you can understand scholarship written in other languages, it's not really frowned upon if you can't speak any others fluently. But not everyone speaks English so it's certainly helpful to have another common language. (There are lots of times when French has come in handy as the only common language I've shared with another academic.)

It also helps to be part of a friendly community full of people who want to share knowledge and resources and help each other out - for me there's "Medieval Twitter" where people are constantly asking for advice on how to read and translate a text, even if they're well-established scholars in the field. There are some arguments sometimes (should Latin and other non-English quotations be translated in books and articles? There was also recently a heated discussion about someone who wrote a journal article in Latin for some reason) but generally it's pretty pleasant and everyone is eager to help.

So hopefully that helps explain it a bit. The keys for me were intensive study with lots of reading and translating, which was built into the grad school program. (And there was probably lots of coincidental luck too.)

Hopefully others will also weigh in on their experiences, because like I said, it's definitely going to be different for every field of history. Some historians might never have to learn another language at all.