It's the subsidiary verb (-て)くる, and it's used to describe 彼 is physically coming closer to 香織さん.
This may seem redundant at first because 近づく ("to get closer") also has a similar meaning, but they play different roles. 近づく and 遠ざかる are about the distance from the target (marked with に/から), while (-て)くる and (-て)いく are about the distance from the observer (which may or may not be the same as the target). For example, you can say AさんはBさんに近づいていった, which means A got closer to B and away from the (implicit) observer.
In your case, there is a narrator that describes this scene as a third person, but 香織さん is working as a temporarily observer here. Or you may think there is an invisible narrator (or camera) near 香織さん. See also: What does ～てくる mean here?
(-て)いる has a totally different meaning. It's possible to use (-て)くる and (-て)いる together, like this:
He is (gradually) getting closer to Kaori.