To beat this boss, you'll need to follow some very specific steps. You need to look out for the exploding carrot attack, which is foretold by a dark aura around the rabbit's head. When he finishes the attack, quickly search the area for a carrot stuck in the ground. Use the push move to fire it at Monster Carrot (this takes pretty good aim). This stuns him and leaves him open to either Power Pole or energy attacks.
When he does recover, he'll revert to the car form. Now just repeat the steps above. Keep in mind that the car gets faster with each appearance. Three successful stuns should put Monster Carrot down. Watch the scene that follows and you'll open both subchapter 5-5 and chapter 6-1. Check the appropriate section for info on 5-5, or skip it and go right to 6-1. Before you move on, however, visit the shop and buy Bulma's new special weapon the rocket launcher. Equip it if you wish and move on.