Some materials (namely those of the Pt-group) would be returned to earth. Others would remain in space for application to structure or as fuel or as water.
The idea of mining asteroids and the moon has been around for decades, and there are numerous symposia and reports on the subject.
One wonders why mining sites of asteroid impacts doesn't get quite so much attention. It would also be interesting to see where the Moon factors into this, it's a lot closer and also has minerals rare on Earth. Though public campaigns against Moon mining would probably be greater than that of asteroids.What I understood from the press conference is that the mining will concentrate mostly on (a) water, which is intended to stay in space, and (b) minerals that are extremely rare on Earth but presumed frequent on asteroids, i.e. platinum; for the latter, it was mentioned that it is one of the few materials which cost more than the price to put them in orbit. It doesn't sound like large volumes need to be brought back to Earth for the business to be profitable.
Edit: Astronuc just got ahead of me. :)