Can I say 'Why are you into studying music?' instead of 'How did(do) you happen to study english?'
Is the expression 'Why are you into ~~" grammatically right?
It's grammatically fine, but that phrase "into ~" is typically used for hobbies and interests, not their work or primary field of study.
I'd also use "How did you get into ~?" instead.
Yes, the expression to be into something, like "I'm into bowling" or "Why are you into baseball?" is grammatically correct.
To be into something means that that thing is an interest of yours. For example, if I said, "I'm into history," that would mean that I have an interest in history and enjoy studying it.
"Why are you into studying ____" does not mean exactly the same thing as "How did you happen to study ____", though. "Why are you into X?" means why does X interest you? "How did you happen to study X" means more simply what events caused you to study X?
For example, if I learn that someone had learned Mandarin Chinese, I might simply say, "How did you happen to study Chinese?" and they might say "My work required me to learn Chinese" or "I wanted to be able to talk to my in-laws." But if I learn that they study languages for fun, I might ask "Why are you into languages?"
+1 to @stangdon. Let me add:
To be "into" something is informal. You might say to a friend, "Hey, I hear you're really into football." But you wouldn't write in an academic paper, "After retiring from the presidency, George Washington returned to his home in Mount Vernon and got really into farming."
"Into" can also be used to indicate entering a profession or similar calling. In this case it is used following a version of the verb "to go". Like, "After college, Mr Jones went into accounting." "George decided to go into the priesthood." This usage can be used in formal writing.