How do I remove a bottom bracket?
Look at the thread on the exposed cup. That tells you which way you need to turn the cup to get it out (there are multiple standards here, and one of the key differences is that some use left hand threads instead of the usual right hand). It looks to me as though the top of that cop should rotate to the right to remove it, in which case it is a left hand thread.

You should be able to fit a spanner onto the raised part on the cup nearest the camera in that photo. Preferably a large spanner with nice sharp edges on the jaws, since the flats are not very deep.

The other side possibly has a couple of holes rather than more flats like this side. In which case you can use a pin spanner or a hammer and chisel to tap it out. Or, as Sheldon suggests, just put a big bolt through and use that to turn the cup.