The idea is to eliminate a lot of the repetitive gear-shuffling that players must do to move armour sets off of idle and wounded soldiers and onto soldiers about to deploy into combat. This is especially common in the early game and when new tech is unlocked, before you have the means to craft matching armour for everyone in a base.
How it works is simple. When you go to the vessel screen and assign armour to your soldiers the game displays a list of what is available, normally armour that is in use does not appear on this list, but with this option enabled any equipment worn by idle soldiers can be selected (idle meaning anyone not assigned to a vessel). If you select an armour that is already in use, the soldier on the vessel and the idle soldier swap their outfits with each other. The swapping mechanic ensures that the idle soldier is left wearing your second best equipment instead of defaulting to nothing.
