Assigning soldiers to an aircraft is not sufficient to send them to the battle. None of them would like to fight the aliens with their bare knuckles. Give them a well though range of weapons and support tools for them to be the most effective on the battlefield. Once you have assigned soldiers to an aircraft, click on Equip soldiers next to the aircraft view to enter the team equipment screen.
Note: Even if you own a base without hangars/ships, you'll be prompted to equip the present soldiers if the base is attacked.
Select a soldier on the top left window. You will see its current equipment on the top right section, informations about his/her capacities or data about the selected equipment on the bottom left part and the magazine on the bottom right part.