As long as she's been in Game of Thrones, Brienne has seen herself as a loyal guard to someone — first, King Renly Baratheon, then Catelyn Stark, and then her Stark daughters. But what she doesn't realize is how she's become a leader in her own right; she's a leader that has made a fumbling squire into a brave knight (we see you, popstar-in-the-making, Podrick), who inspires respect in all who fight against her, and who manages to bring the arrogant Jaime Lannister to heel more than once.
In episode two of season eight, Jaime tells Brienne that he would be honored to serve under her command, which I choose to interpret in all the ways possible. But mostly, it's Jaime showing how much he respects and trusts Brienne as his commanding officer. The gesture immediately overwhelms her because she's always been the one in awe over him, and now the shoe is on the other foot.
