why would my chain slip on the road, but not when the bike is on a stand?
You likely have a problem with worn drivetrain components. You only really notice the jumping under load is my experience. 

It's either a worn chain or a worn cassette in most cases. If you replace the cassette, replace the chain too.

Surprisingly enough, a worn chain and a worn cassette may work well together till one of the two parts are replaced.