It's going to be a lot of work. If the cabinet is one you like, it's worth the effort. Otherwise, I might trash it and find another cabinet.
First, you have to remove the existing board. Chances are it's just glued in. Going on the assumption that it's mostly glued, turn the table upside down to get to the piece. From the bottom, you can see where the side pieces and the bottom piece meet. Take a razor blade cutter of some sort (utility knife, paint scraper, etc.) and cut between the two pieces of wood. (Be sure and measure the piece that needs replacing before you start cutting, down to the 1/4 inch.) Cut that glue joint loose on both sides. Now, stand the cabinet upright and cut it loose from the top. (A paint scraper might work best here, as it's flatter.) Keep working on the glue joint until you get the piece to turn loose. Then, replace the bottom piece with a new piece of wood.
If you have a friend with woodworking tools, you can just cut that bottom piece off flush with the sides and use a router to remove the small amount that would still be there.
I've made a crude drawing of how that piece is in there so you can see how to remove it. (If you run into a staple or a nail, just pull it out.)
If you find it's worth the effort, the replacement will take just gluing it back in and compressing (holding it together and tight) the whole thing while the glue dries.
bk