How can I get the cup stains out of this table?

Wipe off and buff.
For stubborn cloudiness
Try a small amount of white toothpaste (not gel) mixed with baking soda.
Rub lightly with a soft cloth following the grain.
Wipe clean immediately and polish.
Finish with furniture polish or wax
This evens out the sheen and protects the surface.

Avoid:

High heat directly on the table
Abrasive scrubbers
Acetone or strong solvents unless you plan to refinish it

If the finish looks cracked, sticky, or the rings don’t change at all, the coating may be permanently damaged and need refinishing.