I think it is showing what you are saved as in the other persons phone. There are ones where the guy presumably saved a girls number but she didn’t bother to save his and another where the girl saved him as do not answer. I think the point is so you can see how the other person views you. These are mostly joke features and the feature you mentioned is pretty standard now.
The picture is a screenshot of a recent viral video where a woman was being street interviewed and she referred to giving a sloppy bj. She said “hawk tuah” as the onomatopoeia for spitting and it became a a meme. In this picture they replaced the “tuah” with “tui” for text user interface. Hawk is a kind of bird. So the meme reads “hawk tui” is the name of their bird based text interface.