"Go off" can be used in a couple of ways, but I believe the most common are:
When the timer runs out the bomb will 'go off'.
She decided to 'go off' and get a drink.
"Go off" can mean 'to activate' or 'to turn on' such as a timer or a bomb. In "My father is a drinker and one night he chose to 'go off' crazier than usual" it means the father acted like a bomb. He 'exploded' with emotions.
"Go off" can also mean 'to leave'