There are two common ways to harvest sponge: by hooking or cutting. 

Hooking

On a calm day when the water is clear, fishermen work from small boats scan the water bottom until they find a sponge. A long pole with a hook is used to tear the sponge away from its bottom. The hook with standard size can also be used to measure the sponge to ensure it is legal size to harvest.

Cutting

Sponges in deeper water can be harvest by divers who wear diving gear and cut the sponge free with a sharp knife. Sponges are often cut with an inch or so left so that they can grow back.  The sponge is often squeezed to release its spores back to repopulate. 

Sources: 

The Biology and Fishery of Florida's Commercial Sponges 

Apalachicola Sponge Company