Socially, bragging is a very direct attempt to elevate one’s status in a group. If a person brags about his skills or wealth or famous parents, everyone will know about it, and therefore (so he might hope) regard him better. Of course, this is regarded as impolite in American society. People like social status a lot; hence, if left to their own devices, they would brag a lot; so, we have a special rule to punish bragging, to prevent people from doing it out of self-interest.
But why? The obvious reason is that people don’t like social hierarchies. It’s unpleasant to be with people who are always fighting for status. Such fights are costly, and moreover, they compel you to fight for status yourself, lest you be surpassed by everyone else and forgotten.
However, people being people, there will still be a status hierarchy even with a “no bragging” rule. It will just be determined by things people know without being told: race, gender, height, weight, attractiveness, charm, style. Who wants to have those be the paramount considerations? Not very many. Yet, we also don’t want to live in a world with constant status fights. I would much prefer plutocracy to rule by Britney Spears and Paris Hilton, but I also wouldn’t want to spend my days arguing about who has more money. Maybe we ought to have ratemystatus.com, where people can post stuff like salaries and which college they went to, so other people know about it without always having to bring it up?
The Rationalist Conspiracy 
LinkedIn, Facebook, and dating sites do implement this to a large extent.
ratemystatus.com is a good idea, but the need to counter-signal makes it difficult.
I think people find value in having ambiguous status. There can be 5 people in a group and if aggregate status isn’t spelled out all of them can think that they are the highest or second highest status members of the group, even with perfect information of status markers.
Status is an aggregate of multiple different factors. Paris Hilton likely thinks that having people talk about you in the most important part of status and doesn’t care about the size of a person’s rockets, Elon Musk likely has a different view. People can very easily have differing beliefs about how status is being weighed.