There may be a small amount of "performance practice fad" about that, but for the most part it does serve a purpose.
Breath is used in many styles of music as a cue. If you think about wind instrument players, for example, every phrase is preceded by a breath, and experienced players will take that breath in rhythm. As a rhythmic gesture, it can be used to communicate time between an ensemble of players.
String instrument players don't need to breathe to play, but they still do it for reasons of cuing/communication. Chamber music is often heavily involved in this kind of dynamic cuing, where it's not so much a single conductor leading the ensemble, but the ensemble players showing each other how and when they're going to play a certain phrase, when that phrase occurs in tandem with another player.
Now, I mentioned how as well as when, and that's because the nature of the breath can also communicate the way something will be played. The particular breath you referenced sounded a bit like a nostrils-flared angsty-breath, which would not be out of place given the music that they are playing, though whether it was slightly overdone in that situation is a matter of opinion.
Lastly, the more meticulous of us out there will say that you never want to breathe audibly in a recording scenario, because it'll be picked up by the mics, and if you're a wind player after all you should be breathing through your mouth (in fact audible breathing can be used as a diagnosis for excess neck tension in wind instrumentalists). But, frankly, it's become a bit of an idiom in classical string performance, and audiences like to be wowed by how "emotional" a performance is even if they have no idea how to perceive emotion in classical music.
Personally, I don't find myself bothered by it, as it only tends to fill otherwise empty space instead of interrupting a musical gesture.