The phrase is often used in situations of failure or distress, implying sensitivity to others' misfortunes. After the team lost, the coach decided not to criticize them further, to avoid kicking them when they were down.
Pro Tip 2/3
Emotional Impact
Using this phrase suggests empathy and consideration for the feelings of the person in trouble. I heard Ben's business failed, so I didn't ask for my loan back—I didn't want to kick him when he was down.
Pro Tip 3/3
Metaphorical Use
It's a metaphor; it doesn't mean physically kicking someone, but rather adding to someone's troubles. When she gossiped about Lisa's divorce, it felt like she was kicking her when she was down.