What is the purpose of the custom where the bride purchases a tallit for the groom?

Shabbat Shalom, Several reasons have been given for this custom, including the following: 

1. Rabbi Zeiff Wolf of Zhitomir, of blessed memory, a student of the Maggid of Mezritch, of blessed memory, explained in his book Or HaMeir (Sukkot sermon, folio 238b) based on what is seen in Masechet Menachot (149a), which warns regarding the mitzvah of tzitzit as protection from the sin of adultery. It can be hinted at by the verse (Proverbs 6:32): "He who commits adultery lacks sense," implying the absence of the "heart" thread in the tzitzit of one who is not careful with the mitzvah. Thus, on his wedding day and the day of his joy when he enters under the chuppah with his bride, she sends him a tallit with four tzitzit and twenty-eight threads, hinting to him: "Here is your portion given from Hashem, a wise woman giving a heart to rejoice in his portion specifically." 

2. The Shdei Chemed, Part 7 (Section: Groom and Bride, paragraph 11), brought from the Yafe Lev that the Ashkenazi custom is for the bride to send a tallit to the groom on his wedding day, offering a reason based on what the Magen Avraham wrote (Orach Chaim 8:63) that a bachelor, even if he is a Torah scholar, would not cover his head with a tallit. Since she causes him to sanctify himself with this tallit, they established that the tallit is hers. This custom is mainly prevalent among Ashkenazim (which aligns well with the second reason, as it does not apply to Sephardim, who wrap themselves in a tallit even before marriage). However, it did not become widespread among Sephardim except recently (Yalkut Yosef, Sheva Brachot, Part 1, Chapter 6, Paragraph 11). The custom is to use the tallit from the wedding onward. If he already has a tallit, he may designate one for weekday use and the other for Shabbat. 

With blessing, Hillel Meirs