The phrase is informal and used mainly in casual conversation. At the party, Tom drank too much and made an exhibition of himself.
Pro Tip 2/3
Negative Implication
The phrase suggests the attention gained is unwanted and reflects poorly on a person. Running around the office in a superhero cape, Joe certainly made an exhibition of himself.
Pro Tip 3/3
Not Complimentary
It isn't a compliment; it means someone is acting foolishly in public. She sang loudly on the bus, making an exhibition of herself.