How do I remove this metal grating-type thing from a coffee table?

Picture 2 shows it's held on to the legs by a pretty crappy weld. My first thought is "hit it with a hammer" because that weld should be reasonably easy to break; but you run a risk for collateral damage to the leg (bending or twisting) or the rest of the table, that you want to keep.

The proper and clean way to remove that would be with a Dremel or an angle grinder. That way you can target the weld and grind the leg smooth again after it's removed, but you'll need to repaint at least that bare-metal area when you're done, and possibly the whole leg if you have trouble matching paint.

I agree with what ScienceGeyser said though, this bottom grate could be structural in a pretty heavy table and removing it might cause it to be incredibly unstable.