As with all the mitzvos pertaining to the korban Pesach and to Passover night, the reason for this is to celebrate the miracles that God did for the Jews when He took them out of Egypt. In this instance, it is to emulate royalty. Important people eat in their palaces, surrounded by their entourages and retinues. They don’t take their meals “to go” or send out doggie bags.
The Talmud (Pesachim 85b) compares the prohibition to remove the korban Pesach to the prohibition of carrying something from a private to a public domain (or vice versa) on Shabbos: the person is not liable for taking it out until he puts it down. (That doesn’t mean that it’s permitted to remove it without putting it down, just that the person is not culpable until he does so.)
This mitzvah applies to both men and women while the Temple is standing. It is #123 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and is discussed in the Talmud in Pesachim, pages 85-86.