The biggest thing that Mars One has accomplished is getting people to think about going to Mars. That's actually a difficult process, and so far their efforts have done much towards changing the perspective of people in thinking that way. People talk about Mars One in a way that going to Mars hasn't been talked about much. SpaceX, while it isn't a secret at all that they want to go to Mars, hasn't really hit the public perception like Mars One has. In a way, movies like The Martian owe their existence, in some part, to Mars One, for sparking the public interest in traveling to Mars by humanizing it, by picking astronauts.
The Mars One plan, so far as I can tell, is to raise money and purchase the equipment, not to really build things themselves. They have done some whitepaper development, and funded some research. I can't tell the exact relationship, but I believe Mars One is funding research to determining what kinds of crops can grow on Mars. The below text is the evidence that I have for such. It seems at least one of the advisers on Mars One is researching this.
Professor Leo Marcelis of Wageningen University is advisor in the ‘Mars One’ project and one of Wieger Wamelink’s colleagues. He is looking into cultivation systems that should make growing vegetables on Mars possible.
Bottom line, they did a great job at PR for a Mars trip, raising some money, writing some whitepapers, and possibly looking into growing crops on Mars, but not much else. They've pretty much faded away since then as well.
