What does “Why does he get it in for me?” mean?

"Why does that person become so negative toward me", is what the sentence means.

get is a way to say make to have (obtain), and to "have it in for someone" means to have a negative opinion and/or intention toward a person (and it shows through negative things that person does toward the person). It is a form of punishment, in a way. The person has it in for the other because they want to punish them for something (often something that makes no sense except to the person who is doing the punishing).  They think they are teaching a lesson.  No lesson is learned by the victim if the victim has no clue why they are being targeted for punishment.

It isn't a very common thing to say, "to get it in for someone".  The meaning is pretty obvious though, that they are saying he develops that "have it in for me" attitude.

What you wrote is a distortion of the way that it should be said.

Use the verb "have".

But then, you may be British.

As an American, I don't like the way that Brits use "got" in some cases.

"Got" traces back to the verb "get" or the verb infinitive "to get".

Here is a use that I don't like:

I've got a pen. 

If you can trace this verb back to "get", then you should say:

I've gotten a pen.

If you merely have a pen with you, then the verb should be "have" or "to have".

I have a pen.

Mt point?

"Got" is not a conjugation of "have".

I'm more familiar with "why does s/he have it in for me" which means why is someone wishing me ill or desirous of taking vengeance on me.

I have never heard this before. I think possibly it was meant to be "Why does he have it in for me?" Which is a slang way of asking why someone is causing another person upset.

More usually: 'why does he have it in for me?'.   It means he is always looking for opportunities to make life difficult for you.

'get it in' usually means to be on the lookout for opportunity to do evil