You can find good sources about this NASA project on the internet. I just saw one recently, cannot remember where.
If I remember correctly, they were working on several related issues. The first issue was rendezvous with an uncooperative target, which essentially meant a satellite that was dead and no longer under control. That model that is mounted on the wall in one of the photos was a model of their theoretical test target in orbit. If I remember correctly, it is a dead comsat (Update: might be a dead weather satellite). (I seem to remember that they said that they had done a survey of the best possible test targets and unfortunately the best one was then owned by a French company, so it was not something they could count on. But this was three years ago, and they may have picked an entirely different target by now.)
I think that what they were doing was working out rendezvous software connected to imaging equipment. So they had (again, I'm trying to remember) a camera mounted on a robotic arm and that arm was programmed to act like it was a satellite. And then it was supposed to acquire the model on the wall, figure out which way the model spacecraft was oriented, and then maneuver itself up to rendezvous with it. Naturally you would assume that the target might actually be rotating, and this thing was fixed, so I don't know how they accounted for that. Possibly they had a software patch that tricked the primary program into thinking that the satellite was rotating. Dunno.
The screens were them working on one of these rendezvous projects.
The other major things they were working on was a system for capturing the target considering that the target was never built with grapples or anything like that, unlike Hubble or some other spacecraft in the 1980s. I don't know the particulars, but one of the problems is that there is no guarantee that you know EXACTLY what the satellite is even if you have the manufacturer's blueprints in front of you. That's because often changes are introduced during manufacturing that are not recorded. So you could fly your servicing satellite up thinking "We will grab hold of this antenna bracket" and then you get there and discover that the bracket doesn't exist, or doesn't have the strength you expected. So I think that they were trying to develop adaptive software that could deal with stuff like that.
Finally there was a really tough issue--how do you service a satellite that was never designed to be serviced? For instance, because many of these satellites use toxic fuels, often what happens is they fill up the tank through a valve, and then they install a permanent cap on the end of the fill-up pipe. It might be welded on or secured in some other way. It was NEVER designed to be opened up again. So how do you then refill the fuel tanks? That was a really tough issue and I think that is what they have already tested on ISS. That involves things like cutting through the side of a satellite, cutting into a fuel line, and installing a plug. Dangerous things to try.
Remember, the big problem here is that these spacecraft were never designed for servicing, so the question is how do you deal with that?
Major caveat here: I simply got a tour of their facility three years ago when I was there with some VIPs and I am not an expert on any of this stuff. I could be wrong about multiple facts. If you are really interested in the subject, you'll have to do your own research.