The cartoon's title is a play-on-words of the famous phrase to be "mad as a March hare", the origins of which are disputed.
This cartoon begins with Marvin the Martian observing the planet Earth from Mars through a telescope. He is examining a rocket launch that is taking place. As he watches, the rocket takes off from Earth and soon appears to be heading straight towards him. Soon enough, the rocket lands on Mars, and a reluctant Bugs Bunny exits it. It is quickly apparent that he is the only occupant and he has been lured onto the rocket by a carrot and then sent to Mars as what Earth considered an expendable “astro-rabbit.”
Bugs immediately claims Mars as his own (using a metal carrot with a flag inside), in the name of Earth. However Marvin does not agree with this and decides he will not allow Bugs to take his planet away from him. After failing to disintegrate him with his disintegrating pistol (which resulted in Marvin getting disintegrated himself and going off to be re-integrated, "Being disintegrated makes me very angry! [huff, puff] Very angry indeed!"), Marvin gets his time-projector gun and sets it to revert Bugs forward into time so that he can use him as a lifelong slave. However, when Marvin zaps Bugs (though Bugs tried to beg for mercy, which of course makes Marvin angry), he realizes too late that he had the gun in reverse, so Bugs is reverted into a Neanderthal Rabbit, who immediately grabs Marvin and crushes him with just one hand; Marvin goes off to be regenerated again, while saying "Well, back to the old electronic brain!" (a possible reference to Hare-Way To The Stars). Bugs then eats the metal carrot and breaks the fourth wall while speaking to the audience about how Elmer Fudd and all the hunters back on Earth are going to be in for a big surprise due to his Neanderthal form when Bugs gets back to Earth, as the cartoon ends.