What do the Action Points in the equipment section mean?
n_palum is pretty much right about the meaning. It's a feature that resembles standard role-playing games where it counts equipping an item, like a weapon or armor, as an action. Thus, to equip a chestplate with 5AP during battle, it'll require you to have 5AP available to spend during a turn.
This thread backs this up.
I think it's how many AP it costs to equip during battle...?
This exactly. I switched a piece of equipment in battle earlier and realized it cost ap to switch.