I found a perfect answer to this question on Reddit (edited to make it a bit more presentable):
It's for two reasons I know of:
- It completes her character arc. The entire reason she was such a mess at the beginning of the show and the whole reason she needed to learn through Ryuuji how to have basic interactions with other people without being a rude, condescending, violent bitch is because of her wrecked home life and broken upbringing. By the end of the show, she's matured a lot. When Ryuuji inadvertently damages his own home life in the last 2 episodes, Taiga watches him get up and fix it in a matter of days. It only makes sense for Taiga to decide at this point that she has to come full circle: She has to go face the issues that started all of her problems now that she's mature enough to do so. She has to go home and mend her relationship with her mother and step-siblings, and she does.
- Two, because it symbolizes equality in her relationship with Ryuuji. Throughout the show, their relationship is actually rather unhealthy, as Ami points out a number of times in the later episodes. Ryuuji is doting on Taiga and doing a lot of things for her that she should be doing for herself. Their relationship is more of a father/daughter relationship than one of equals (Ami also says this directly once). This is unhealthy for Ryuuji, too, as it feeds into his "I have to take care of everything and everyone" mindset, one he got from his own home life. Taiga leaves to fix her own shit, demonstrating her independence from Ryuuji as a capable person of her own, and Ryuuji accepts and respects her decision despite it hurting him, demonstrating that he acknowledges Taiga is an equal to him and not someone he needs to do everything for.
The scene after the end credits, which you've hopefully watched by now, is almost between two new people. It's one of mature, equal people who truly understand each other.
It is never explicitly mentioned in the Light Novel or the anime as to what exactly she did during this time period. However, her note to Ryuuji:
As translated in this answer reads:
Ryuuji, I... always thought that there was no way anybody would ever love me. But maybe I was just running away [from my problems?]. I blamed my lack of confidence on my parents... on my situation. But even so, you loved me the way I was. I'll be proud of myself...
Because I want to love you, Ryuuji.
Which does imply that she went to spend time with her family to change herself and become a self-dependent person as stated on Reddit.