Impassibility means that God has no nerves, blood, or flesh. So he doesn’t get hangry, mad due to lack of sleep, or dopey. He is Spirit (John 4:24). So he doesn’t have emotions that fluctuate on the basis of hunger or lack of sleep.
He is not like the Greek and Roman gods whose vices are ever so human. They commit adultery, become enraged, and so on.
God is simply not like that. He is not like the other gods, as Isaiah tells us. He knows the end from the beginning; he created all; he has no visible form.
So Scripture accommodates itself to us by saying God has an arm, nostrils, is fire, is like a bird, etc. These tell us true things about God that fit our created capacity.
The alternative would be a book that feels like abstract philosophy about God’s invisible nature. God instead delights to communicate with us according to our capacity. We do the same with our children.
In the end, impassibility means that God is not a Greek god who will betray you on a whim. He doesn’t get mad when he doesn’t get his way. He doesn’t let things ruffle his feathers.
When God says, “I love you,” nothing can prevent that love from coming to you. It is stable, forever. It’s not based on a sugar high, or a feeling of elation that feeds. God has no hormones to produce such an effect.
When he acts, it’s all of what he is that acts. He is what he is. And that’s really good news.