And what happens after the satellite is captured?

The article doesn't say. I suspect that the capture craft would then have to deorbit itself and the captured sat which makes it a multi-billion dollar waste. If it can at least capture several large objects prior to deorbiting it might not be such a terrible waste.

I expect the targets will be tumbling. What happens when you spear a big object tumbling? The deOrbit vehicle winds up (literally!) spinning on the end of its tether like the rock in David's sling. Now what? Maybe it's still possible to apply thrust, but I like to know how the developers intend to deal with the problem.

Instead of harpooning it, punch it out of orbit down toward the earth, let the denser air disintegrate them. If you miss, the shot would still direct earthward and also have to disintegrate.
Or maybe just the orbital tugboat approach, and nudge it towards a burn.
Either way your clean-up machine still needs to get up, and be able to move around, and maybe it can even feed off of the scrap.

How about micro-sats with a thick, sticky pad that can conform to the target? Give the micro-sats a control system with enough RCS fuel to aim and hold the target to point a small solid rocket motor retrograde.
Once the thing reaches the right point in orbit so it'll splash instead of crash, fire the rocket then wait.
Sticking it onto a point of balance so the rocket won't set it spinning would be a *ahem* sticking point.
An alternate method would be to have the sticky part on the end of a tether then unreel it prograde from the target so that when the rocket fires any twist of the target being pulled off center won't have much effect. A possible problem there would be ripping pieces off the target.
What has to be done is to just lower the perigee enough so that the atmosphere will finish the job of bringing the debris down.

In theory once you harpoon a satellite and bring it to a much lower altitude you could use the tether to finish deorbiting the prey and fling the hunter back up to use an additional harpoon on further prey.

All that debris has kinetic energy. E= 0.5 m v^2 Why not harvest that too? Could speed up the catcher (higher orbit?)

@ Rob Tillaart Too avoid generating more pieces of debris you have to match the velocity of the piece you are capturing pretty closely so there won't be the energy differential to work with.

Rather than sending stuff out of orbit piecemeal, keep the pieces together in a relatively more manageable aggregated unit that could then be de-orbited to a preassigned surface area. Should such an area be the Russian steppes, the Australian outback or USA desert, the salvage value would go considerably far in recouping the expense of the harvest. Incorporate a retro-rocket module with parachutes on a tether, and the hazard below would be diminished, not to mention how the debris could be inspected. The effort should pay off in research, that said debris would reveal the failures (or success) of the past. Beyond answering open questions about past performance, the salvaged hi-tech material would contribute to future missions.

@ anthodyd With the exception of titanium spheres most of the stuff will burn up during reentry.