Mr. Klipspringer exists in The Great Gatsby as a kind of chaperone. Nick is in a delicate position. He has agreed to let Gatsby use his cottage as a place to encounter Daisy. Then Gatsby invites them both to his conveniently situated mansion to "show them around." Nick likes to think of himself as a respectable sort of person. For example, he is too moralistic to consider developing a "connection" with Meyer Wolfsheim. But he has a moral responsibility to Daisy--at least in the 1920s. He is being put in the position of a pandar. His creator, F. Scott Fitzgerald, wants to make Nick's actions look as innocent as possible, although what is actually happening is that Gatsby is using Nick to meet Daisy so that he can get Daisy over to his mansion and ultimately into bed. The book was published in 1925. Things like that were not described in any detail because of the unwritten publishers' code. Although there are two adulterous affairs going on in the book, one between Tom and Myrtle and the other between Gatsby and Daisy, there are no depictions of any physical intimacy--unless you can count Tom breaking Myrtle's nose. So Mr. Klipspringer is a sort of stand-in for Nick. When Nick leaves, he is not really leaving Daisy "alone" with Gatsby, with all that implies. After all, Mr. Klipspringer is there! When Nick leaves he can feel that he has behaved like a proper gentleman. He knows Daisy and Gatsby would love to get rid of him, just as Gatsby will undoubtedly get rid of Klipspringer when Nick is gone. Klipspringer is only created out of thin air and brought into the scene to show that there is somebody else in the house.
They had forgotten me, but Daisy glanced up and held out her hand. Gatsby didn't know me now at all. I looked once more at them and they looked back at me, remotely possessed by intense life. Then I went out of the room and down the marble steps into the rain, leaving them there together.
When Daisy holds out her hand, the gesture says a lot of things. She is thanking Nick for bringing her and Gatsby together. She is thanking him for having the savoir faire to leave them there alone. She is showing him that she is perfectly safe and happy to be where she is, and that it is all right for him to drop his role as escort, chaperone, host, cousin, male guardian--or whatever he was supposed to be.