The Genovese (among most Italians) really appreciate fresh food - apples that don't come wrapped in plastic, fish that isn't sealed in styrofoam containers. And so they shop daily for produce that's in season, and fish that has come off a boat that very morning. And with the layout of Genoa as it is, historically there really were no central squares in which to set up small markets. Thus, a huge central market was created instead, just outside the city walls.
If you come on the weekend, be prepared for massive crowds. For this reason, I love coming on Monday afternoon - very few people means you can really explore the stalls and have your pick over the freshest green tomatoes or the sharpest piece of pecorino cheese.