It’s time to translate more terms. Here, the Torah says a “toshav” and a “sachir” may not eat from the korban Pesach. What are a toshav and a sachir? A toshav is what we call a ger toshav, a non-Jew who lives in Israel and obeys the seven universal (Noachide) laws. Basically, non-Jews are not required to become Jewish; all the Torah requires from them is to observe a basic catalog of laws (such as not to kill, steal, or worship idols, among others). However, this does not entitle them to eat from the Passover offering.
A sachir is a non-Jewish hired laborer. He may live in the Jew’s home or the Jew may be responsible to feed him as part of his wages but, again, this does not entitle him to eat from the korban Pesach. This may seem obvious, but contrast it with the previous verse. 12:44 says, “A (non-Jewish) slave, purchased with money, must be circumcised, then he can eat from it,” (“it” being the korban Pesach). So, since a non-Jewish slave is treated like a member of one’s own household, the Torah clarifies that the contractor who’s going home after the job is finished is not included.
Another explanation of sachir is a non-Jew undergoing conversion, who has already been circumcised but has not yet been ritually immersed in a mikvah. See Talmud Yevamos 71a and the Sefer HaChinuch on this mitzvah.
The reason for this mitzvah is similar to that of the previous one: the korban Pesach celebrates the relationship between God and the Jewish people. It’s “members only.” Having one foot in the camp isn’t enough to participate in this particular ritual. If the toshav or the sachir wants to eat of it, they have to take the plunge (literally) and actually join the Jewish people.
This mitzvah applies to both men and women while the Temple is standing. It can be found in chapter 9 of the Rambam’s Hilchos Korban Pesach and on page 71a of Yevamos. It is #126 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos.