How do 6 legs give a bee more stability and functionality than just 4 would give?

At any time moving around the hive at any angle, including completely upside down, the bee has a minimum of 3 legs in contact with a surface at any time giving maximum stability in any position a bee takes while completing her tasks. Any pair of legs can be used for specialized tasks while the bee’s body is secured by the tarsal hooks of at least 3 legs still in contact with the surface or hanging onto other bees’ bodies, as when grouping to make wax comb, sometimes called “festooning.”

Diagram of comb building workers is taken from The Beekeeper's Handbook, 5th ed., Sammataro & Avitabile.