Why does limestone fizz in vinegar?

Because acid takes it apart.

Calcium carbonate cannot survive contact with acid. The acid breaks it down, and one of the pieces that comes away is carbon dioxide. That gas is the fizz.

Geologists rely on this. A drop of weak acid on an unknown rock, and if it bubbles, it is limestone.

Worth knowing
The same reaction explains two very different things. It is why lime corrects sour soil, cancelling the acid so plants can feed again. And it is why acid rain slowly eats away limestone buildings and carvings. One reaction, useful in a field and destructive on a wall.

Source: Royal Society of Chemistry.