It says in the Rambam that there is a custom not to say Tachanun at Mincha on Erev Shabbos and Yom Tov. What is the reason for this custom?
Taamei HaMinhagim, entry 128, says that we do not say Tachanun on the eve of Shabbat or Yom Tov (and other eves), as a reminder to pray the special prayers in that evening's prayers, or at least as a reminder that the special day that starts that evening.
We do not say Tachanun on Shabbos and Yom Tov which are "Yemei Simcha" days of Simcha. Since Mincha prior to these days we are already in the Simcha mode, we do not say Tachanun either.
The Gra (Biur Hagra to OC 267:1) ties this to the rule that mourners can stop certain rules of Shiva starting from Mincha time on Friday in preparation for Shabbat, as outlined on Moed Katan 27a (also, Rambam Avel 10:2).
I note another instance where we see that the afternoon preceding a day is tied to the following day, namely the rule that only on the 7th day of Sukkot in the afternoon can one remove his belongings from his Sukkah and move them inside in preparation for Shemini Atzeret as outlined in Shulchan Aruch OC 666.