If the oaths exchanged between Sansa and Brienne sounded familiar, that's because it is the exact promise Brienne made to Sansa's mother in season two. The emotional response both Brienne and Sansa had to this solemn vow was heartbreaking to watch.

Without hesitation, Brienne lowered her sword and swore a vow to Sansa. "Lady Sansa, I offer my services once again," she said. "I will shield your back, and keep your counsel, and give my life for yours if need be. I swear it by the old gods and the new." 

Brienne has wanted nothing more than to fulfill her promise to Catelyn of finding and protecting her daughters. Both Arya and Sansa eluded her grasp for a long time, but now she is in her rightful place.