The word "assess" means to check or test. "Assessing the damage" to something means figuring out how badly it was damaged. 

You "assess the damage" in situations like these:

• You assess the damage to the body of someone who's been injured.
• The government assesses the damage to homes and businesses when there's been a natural disaster.
• An insurance company representative assesses the damage to your car when you've gotten in a car accident.