The Talmud (Megillah 5b) explains each of the different expressions for this holiday to mean a different method for celebrating the day. Simcha (“joy”) is interpreted as not giving eulogies (in the event of a death); v’mishteh (“and feasting”) is interpreted as prohibiting fasting; and v’yom tov (“and the holiday”) is interpreted as prohibiting work on Purim. Later, the Talmud (Megillah 7a) interprets the phrase mishloach manos (“sending gifts”) as the requirement to send through a messenger at least two kinds of food to at least one friend.
The Rambam (Mishneh Torah, Hilchos Megillah 2:15) writes that even two poor people are required to send another poor person some food to fulfill their obligations.
The Trumas HaDeshen writes that the exchange of food is intended to make sure everyone has enough for the feast.
M’nos HaLevi writes that we send gifts to promote friendship because achdus (“unity”) rescued the Jews.
The Midrash HaGadol on Devarim points out that this demonstrates the greatness of chesed because we were rescued because of it.
Ginzei HaMelech writes that we use a messenger because this shows achdus (“unity”) in requiring another person to get involved in this mitzva. Similarly, he points out, this is why Megillas Esther always uses Yehudim for Jews, since the root of that word is echad, one. Furthermore, the giving of gifts through messengers acts as an additional tikkun for Yaakov’s giving gifts (Bireishis 32:14-17) to Eisav, the ancestor of Amalek, through messengers.
The Vilna Gaon and Midrash Shmuel note that the Jews’ celebrating in this way parallels the three parts of Haman’s plan (Esther 3:13); the joy serves to counteract Haman’s plan to destroy the Jews, the feasting serves to counteract Haman’s plan to kill the Jews, and the Yom Tov serves to counteract Haman’s plan to annihilate the Jews.
The Vilna Gaon writes that, eventually, Purim was not accepted as a full Yom Tov because that would keep people from performing the other mitzvos of Purim.