Depends where it hits.

On the tether there is a "zero" point. Below that point things fall toward the Earth, above that point things "fall" away from the Earth.

So it depends on where it broke and how much weight there is above vs below that point. (To keep the tether in the air they put lots of weight above the point, which falls away from the Earth and pulls the tether taught.)

If I were making one of these I'd make several tethers, spaced such that a single object can not break more than one at once. Then tie the tethers together periodically.

This object is about max: https://en.wikipedia.org/wiki/Tunguska_event so I'd make the multiple tethers at least 200m apart. That's pretty far apart, so I'd invest in better detection technology to reduce the size maximum undetected object. (Although that raises the question of what to do if you do find a big one and it's headed right toward you.)

Your question's been well answered by ars, but often the next question is: how dangerous would it be to people on earth if it broke? The answer is not at all. As lutorm points out the cable would be vanishingly thin - there would be a lot of gentle drifting and no firey death from above.

I didn't see it mentioned in this article, but the cable is actually very small. I read this older NASA study that said that for a carbon nanotube tether, an optimal cable would be a thin tape something like an inch across. That has pretty small cross section for being hit by a meteoroid.