Translation:I will get off at the next station.
Agreed. The Japanese is specific about it being a station, but 'stop' isn't specifically a station. However, I can only see this being used while travelling on a train, and in English I would generally use 'stop' in that context. (I'd probably say "I'm getting off at the next stop", too, but that's just another can of tense worms!)
That doesn't change the fact that your English needs to be grammatically correct. Don't get angry at me because your English was wrong. You asked why it was marked incorrect and I gave you the answer. I didn't do it rudely, so you had no reason to be rude or defensive. "I get off at the next station" might be colloquially understood, but it is not grammatically correct. "we" cannot be used because where the subject is left out, it is implied that the subject is the most basic and obvious. If it were a 'we', then you wouldn't leave it out, as you need to provide context.
Even more gets frequently omitted in conversation. Take your last sentence as example and let's modify it:
"Not far, I get off next station."
I know this is true because in my native language this cannot be done as it will just sound and be grammatically off, not just one or the other but both.
All the necessary information is still conveyed and the grammar checks, by not using "at the" we treat the next station as a point in time. This is intuitive and being over explained now.
And the answer is that while に is a place marker, で is more specifcally used to mark an action at a place. If に was used instead... it would be like getting off onto the surface of a train or something like that... consider the following to get kind of an understanding of what I'm trying to say.
紙に書きます (I will write on paper [in the sense that I am making marks on the surface of it, presumably with a pen])
紙で書きます* (I will write on the paper [in the sense that I am standing on top of a piece of paper writing. I might not even be writing on said piece of paper. The paper is just the location where I am doing the action of writing])
With that in mind, I would say that while it is incorrect, people would still understand what you are saying.
*Though if I did hear someone say 紙で書きます, I'm more apt to believe that the speaker wrote by means of paper instead of at the paper