As was the case with the previous mitzvah, the underlying reason for this mitzvah is to celebrate the Exodus and to acknowledge the elevated status the Jews enjoyed when God took them from Egypt. To quote the Sefer HaChinuch on this mitzvah, “It is not (fitting for) the honor of royalty and their advisors to scrape the bones and to break them, like dogs.” In other words, that’s the way that people who are poor and hungry act, not esteemed people of leisure, whose ways we emulate on Passover night. (Even today we copy the ways of royalty at the Seder, such as by reclining.)
The Sefer HaChinuch continues by explaining the reason for the large number of mizvos pertaining to Passover in general and to the korban Pesach in particular. He says that, psychologically speaking, a person’s attitudes are shaped by his actions. The Torah gives us many mitzvos to commemorate the Exodus – and to act like a holy, important nation – in order to reinforce these things, so that the Jewish people will internalize them.
Missionaries use the alleged “fulfillment” of this verse as a prooftext, but it is a commandment, not a prophecy, and it has nothing to do with the messiah.
The laws regarding what bones are included – for example, if a sacrifice becomes impure and is no longer usable – are found in Talmud Pesachim 84a-85a. This mitzvah applies to both men and women while the Temple is standing and is #121 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos.