There are two reasons why having an inflated view of oneself is useful (there are also, of course, reasons why it's not useful, too).

First is that bragging is very widespread technique for winning mates, both in humans and animals. If you can't brag convincingly -- and internal belief certainly helps -- you're falling behind in the competition.

Second is that overestimation of your own abilities leads to feelings of confidence and self-assurance which are useful feelings to have in general (subject to reasonable limits, of course) and, in particular, are very valuable in giving out the right signals to potential mates. In the latter case, though, I think there's a noticeable difference between the sexes -- being self-assured is much more valuable for (straight) males then for (straight) females.