When you do something on a whim, you do it out of a sudden desire. It's not something you've been thinking about for ages, it's an impulse.
A whim is an odd, abrupt desire or idea. When you do something on a whim, you do it just because a sudden thought hits you and you follow it. Maybe you go to the grocery store intending to get pasta and tomato sauce but, on a whim, bring home pancake mix and sausage instead. It's not for any logical reason — you just saw the pancake mix and suddenly wanted breakfast for dinner.