From the mouth of the actor in charge of playing the current Three-Eyed Raven, Isaac Hempstead-Wright:
"He's got this wealth of important information that really needs to get to the right people. And so by the start of season seven, Bran is in many ways a very different character," Hempstead Wright said. "He's the Three-Eyed Raven — he's not Bran Stark, which means he's really just a vehicle for the greater world's fate. That is what Bran's destiny is and what he's doing in season seven."
— Isaac Hempstead-Wright, INSIDER
"Bran is existing in thousands of planes of existence at any one time. So it’s quite difficult for Bran to have any kind of semblance of personality anymore because he’s really like a giant computer."
— Isaac Hempstead-Wright, Entertainment Weekly
Essentially: being in all places at once, throughout time (past, present and potential future) has been a gift and a curse for Bran, as he is no longer the Bran Stark we knew before, even though he says himself that he remembers what it was like to be Bran Stark, a boy from Winterfell, scattering your mind to bear witness to every event ever in the world, as well as being given the burden of the responsibility of seeing the world through the peril of the Night King, seems to have this extreme and literal greater-goal effect on his character. He repeatedly confirms what the actor has also clarified: he is no longer Bran Stark.
His apparent apathy and emotionlessness are a consequence of not having the same perspective that normal people have. We see Meera saying, "My brother died for you", having no effect on Bran because he has watched hundreds of thousands of deaths and births and horrific events, he has a completely shifted perspective, almost god-like, on the relatively smaller events or interactions, due to his omniscience.
Since I first wrote this answer, an official HBO interview with actor Isaac Hempstead-Wright, over at the official behind the scenes blog, Making Game of Thrones, has come out, further confirming what has been said, as well as adding some more insight:
HBO: Do you agree with Meera’s assessment that Bran died in the cave back in Season 6?
Isaac Hempstead Wright: It’s quite a bold thing for Meera to have said. Sadly, I think in many ways she was right. It’s just this whole idea that Bran has become a much smaller part of the character’s brain, when before 100 percent of his head was taken up with being Bran Stark. Now, that’s just one tiny file in a huge system. But certainly, he’s almost completely a different character. He acts utterly differently, and really any semblance of personality he used to have has gone.
HBO: Is there anything left of the old Bran?
Isaac Hempstead Wright: In Episode 4 [“The Spoils of War”] he says, “I remember what it felt like to be Brandon Stark, but I remember so much else now,” which sums up exactly the situation Bran is in. There is a flicker of Bran left in him, but really, can you imagine putting the entire history of the universe, every single moment, every single second that ever existed in one person’s brain? You’d think it would just short circuit. Bran just becomes this calm, zen character. He’s really like a human supercomputer.
HBO: How challenging was it to essentially take on that new character while still maintaining the fact that you are in some small way Bran?
Isaac Hempstead Wright: It was definitely difficult to get it right. I had a meeting with [series creators] David [Benioff] and Dan [Weiss], and they wanted him to be quite monotone and agenda-less, but at the same time have a slight flair, so it wasn’t just like listening to a robot talk. There had to be a sense of mystery and wisdom to him. He was sort of inspired by Dr. Manhattan in The Watchmen series — being in all these places at once, in all these time zones at once. I tried to base it on the old Three-Eyed Raven [played by Max von Sydow in Season 6] and have a sense of this wise, old, man sitting in a tree. At the same time, still have that slight spark somewhere in there where you know this is Bran Stark. It was a fine balance.
—"Isaac Hempstead Wright on the 'Fine Balance' of Playing Bran", Making of Game of Thrones