To my knowledge 是 means "is" and the usage of 的 that I am familiar with is that it is like a possession particle. I don't understand how either character functions in the following sentence:
你是怎么学习汉语的？
Nǐ shì zěnme xuéxí hànyǔ de?
How do you study Chinese?
If I took a blind stab at translating the English to Chinese, I would have went with something like
你怎么学习汉语？
Is this attempt incorrect?
How do I understand the grammar of the original Chinese sentence?
The trick here is that 是...的 is wrapping the phrase 怎么学习汉语 and adds emphasis to the how part of the phrase. The 是...的 is fairly subtle and there are a few decent articles on it online like this one.
As another responder mentioned there is a bit of a past tense sense to the construction because it's implied that you're asking for details about something that happened, so the translation is kind of like "How did you learn mandarin?".
This is one of those questions that a native speaker with a sound grammatical knowledge might be able to answer better, but from my point of view (as a native speaker with little grammatical knowledge), I think the best way to explain would be to show what the difference is to the listener.
As you may or may not know, the particle 是 is not always necessary from a grammatical perspective, especially when there is no ambiguity around the various components of the sentence, so grammatically both
你是怎么学习汉语的？AND 你怎么学习汉语？are fine.
If someone asked you 你是怎么学习汉语的？(i.e. you are not reading this sentence from a page of a book), then it is more likely that they are putting an emphasis on the process in which you have managed to learn/acquire the language.
However, 你怎么学习汉语？can mean one of two things:
In fact, 你是怎么学习汉语的？can also have this second meaning, but this would fall into the category of a very awkward way of asking this question, which is more natural without the 是 if you want to imply this meaning.
The only way to be certain is to ascertain the exact context of the sentence/conversation (as is often the case with some Asian languages).
Example sentences:
有很多不同的语言, 你怎么学习汉语的？ There are many different languages, how come you study Chinese?
有很多不同的语言, 你是怎么学习汉语的？ Probably should be using 为什么 instead of 怎么 when accompanied by 是
你说的很流利, 你是怎么学习汉语的？ You speak fluently, how do you study Chinese?
你是怎么学习汉语的？ implies past tense, namely In what way did you learn Chinese? or In what way have you learned Chinese?
你怎么学习汉语？is a way asking How will you learn Chinese or How do you learn Chinese.
你 means you.
In this example, 是 is used to determine something. 你怎么学习汉语的？ can be complete sentence too.
In this example, 怎么 showed that you ask how to do something(学习汉语).
学习 means learn.
汉语 means Chinese language. In this example, 汉语 is something you want to learn since you asked it in the question and appear after 学习.
你是怎么学习汉语的？AND 你怎么学习汉语？
My take:-
你怎么学习汉语? --- How do you study Chinese?
你是怎么学习汉语的? --- Show me how you study Chinese.
In the latter example, the 是 and the 的 implies asking for more relevant details.
It is oversimplified to think 是 just translates as 'is'. Also, analyzing a chunk of language separated from its context can often lead to misunderstanding. (Ask any politician!)
是： 日 + 正
是： 日 + 止 + 一
的： 白 + 勺
是：(in the light of) the sun 日 right, true 正
是：（in the light of） the sun 日 a foot 止 stops at the line 一 ： right, true
的：(in the light of) the rising sun 白 (clearly see what is in) the spoon 勺
是····的： a double dose of 'rising sun'. Maybe that's where this idea of emphasis comes from!
是非： right and wrong
How do you study Chinese?
你怎么学习汉语？
你如何学习汉语？
Why do you study Chinese?
你为什么学习汉语？
