While the Professor overzealously called it the "perfect pet," it does have a flaw. A BEEBO can only be fed once during its entire lifespan. If it is fed more than that, it will become obsessed with eating and consume everything in sight. It will also grow larger as it eats and eventually explodes and produce offspring.
When the initial BEEBO was created, the Professor gave it to The Powerpuff Girls who promised not to feed it more than once. They did their best to follow the instructions regarding feeding BEEBO, however, they were unaware that the others had already fed it, so each of the girls fed BEEBO once. BEEBO then became obsessed with eating as the Professor said it would, and grew too large for the house.
It started eating whatever and whoever it could find in Townsville, including The Mayor and Ms. Bellum. It even ate The Powerpuff Girls when they tried to stop it. By eating Bubbles, it had consumed enough to cause it to explode and thus reproduce. This also freed everyone who was eaten by BEEBO.
Professor Utonium warned everyone in Townsville about limiting the feeding of a BEEBO. The Mayor, not completely understanding ("once a day," "once in a while"), was not allowed to have a BEEBO, much to everyone else's amusement... that is, except for the Narrator, who points out he didn't get a BEEBO either. This is pointed out before his trademark ending line.