Broggy is fought once in the game; he fights Bowser thinking he is an intruder. Bowser has to attack Broggy with his fire breath, and whilst Broggy's burnt, Bowser can attack his band-aid until he's defeated. Watch out for him tackling right into Bowser, and be aware that the fire will self-extinguish after a couple of turns.
Broggy is available as a special attack after all fifteen Blittys have been saved. This special attack is called "Broggy Bonker", which costs 20 SP.
If Mario and Luigi talk to Broggy, he will growl in a manner much more menacing than he does to Bowser: "BROOF BROOF BROOF!!!!!", and not allow them to buy gear from him.