Do they take a point of damage when first deployed and every "ready action" afterwards or does the damage only happen from the first time you ready them?

It's an action, which is optional. It isn't some kind of forced reaction. It gives you the option to unexhaust/ready the unit (at the cost of dealing it a damage) in an action window in the combat phase. The usual use would be to do the action so as to allow it to attack a second time during a combat round.