As with all the mitzvos pertaining to the korban Pesach, this is to glorify the miracles that God wrought for the Jews when He brought them out of Egypt. As with the previous mitzvah, this one causes the Jews to emulate royalty in recognition of their new, exalted status. Kings don’t eat leftovers, so neither do the Jews when it comes to the Passover sacrifice. (Sorry! No cold korban Pesach sandwiches after Passover!)
This mitzvah is what’s called a “lav she’nitak l’asei,” a prohibition that is remedied by an obligation. In other words, “Don’t leave any meat over but if you do, you must burn it.” (Lashes are not administered for the violation of such a prohibition.)
This mitzvah applied to both men and women in the time of the Temple. It is discussed in the Mechilta on parshas Bo and in the Mishneh Torah in Hilchos Korban Pesach. It is #117 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos.