You have to select the unit then go back over to the left and click the button that confirms the order ("assign", I think, don't have the game on this computer to check). All troop load/unload/drop orders work this way. Not sure why they are different.
I'll add a question of my own: is there a way to select/load multiple units at once for a large ship or fleet? I have discovered a large invasion has a frustrating amount of troop-related clicking.
Yes, IF you have organized your ground troops as it is intended, i.e. four batallions are assigned to a brigade HQ, 4 brigade HQs are assigned to a division HQ. Then you check the "Load Sub-units" or something to that effect checkbox and only order to load, for example, 21st Heavy Assault Division HQ to load the entire division (provided you have enough space)