Q. What is the reason for the strange minhag of saying Purim Torah?

A. Osri Lagefen (p. 183) explains that Purim Torah is a result from the mitzva of “Lebesumei BePuriya” (Shulchan Aruch O.H. 695: 2). And it proves a most important point. Namely, that even when Bnei Yisroel are in a state of inebriation, what they speak is divrei Torah, as opposed to other cultures, Albeit this divrei Torah may be not entirely reliable. He also mentions that some of this Purim Torah could have a strong moral value, when said properly, since words of necessary admonishment that one may not usually say, become then acceptable. Ohel Moshe (p. 210) describes how Purim was kept in the Mirer Yeshiva mentioning that in the presence of Rav Yerucham zt”l, talmidim would recite Purim Torah. Similar accounts are legend from many other yeshivos.
A curious source is quoted in Nitei Gavriel (Purim 30: p. 587). Since on Purim poor people would put on disguises and go around collecting matonos laevyonim, Goyim would see an opportunity to cheat and do the same. As a security check, one collecting would be asked to recite something of Torah, that even a child would know, and it indeed sounded like Purim Torah.
Others mention that the Torah was accepted a second time on Purim. However, this time it was done with willingness and love, reflected on the fun and joy spirit of Purim Torah.
Horav Shlomo Miller Shlit”a added that the “Ad delo yoda” mentioned as the measurement of inebriation in Purim, is indeed the demonstration of what one really knows.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a.