To grow quickly in size or importance, like a snowball rolling down a hill and getting bigger.
When something snowballs, it usually gets bigger and bigger very fast.
Examples:
The problem snowballed and got worse.
Her idea snowballed and soon everyone was talking about it.
The project's success snowballed and soon it was famous.
The snowballing issue became a big problem for everyone.
The rumor snowballed until everyone knew about it.
The excitement snowballed until the whole town was talking about it.