Jaime was in the hot seat in the April 21 episode of Game of Thrones. See, not only is he a Lannister, which is reason enough for Northerners to hate him, but he'd arrived to speak to Daenerys Targaryen, whose dad he kind of sort of murdered in cold blood. While the Kingslayer had come to offer his services, both the Mother of Dragons and Lady Sansa were understandably skeptical. Fortunately, Brienne defended Jaime on Game of Thrones, showing that the Kingslayer has friends in Winterfell, after all.
"I know Ser Jaime," Brienne began. "He is a man of honor. I was his captor once, but when we were both taken prisoner and the men holding us tried to force themselves on me, Ser Jaime defended me — and lost his hand because of it." Now it seems like Brienne is defending him, in turn. She continued, telling Sansa, "Without him, my lady, you would not be alive. He armed me, armored me, and sent me to find you and bring you home because he'd sworn an oath to your mother."
It was a noble, brave thing to say, and Jaime was understandably moved. And Brienne's testimony turned out to be necessary, because it convinced Sansa to allow the Lannister to stay in Winterfell after all.
Audiences will recall that Jaime and Brienne had quite the adventure back in Season 2, when they were both captured by the Boltons. Jaime had previously been the prisoner of Catelyn Stark, and Brienne was escorting the Kingslayer to King's Landing. On the way, however, they were intercepted by some of Lord Bolton's men. The two bonded through this traumatic experience, which Brienne touched on in her plea to Sansa. She said that when men tried to rape her, Jaime stepped in, which resulted in the loss of his hand.
After Brienne defended him, the two were able to talk one-on-one in the courtyard, where he revealed that he's not the fighter he used to be. This is ironic because it's likely because he lost his hand protecting Brienne that he's no longer a capable warrior. Nevertheless, Jaime asked if he'd be able to serve under her command — if she'd have him.
It was strange to see the cocky Kingslayer so thoroughly humbled, but it was clear that he meant every word. Fortunately, Jaime had a chance to repay the favor in Season 8, Episode 2, when he knighted Brienne before their (still upcoming) battle with White Walkers.
The two characters have always had intense chemistry, and while in earlier seasons, it was painfully obvious that Brienne was in love with the Lannister, the feelings might be mutual this time around (but don't tell Tormund, because he's got his wild eyes set on the newly minted knight, too).
It's doubtful that both of these characters will survive this final season, but it's certainly satisfying to see them both vouch for each other before it's all said and done. After all, a Lannister always pays his debts.