Historically, the exchange of boasts has been a form of entertainment for patriarchal societies, as well as a method of influencing or lightly threatening to avoid conflict. Boasting would usually be done in a group environment, at the end of a meal or while celebrating victory. It was considered normal to exaggerate deeds while never lessening the deeds of others; in this way, the men could keep their own self-respect. A good boast could enter the realm of rumor and legend, and turn into accepted truth about a person's reputation.
In Beowulf, there is mention of boasting on several occasions; this is appropriate for the time period, when warriors would gather and tell tales of their heroism. Beowulf himself engages in a mighty boast while dining with the Danes, in order to represent his achievements and to assure the crowd that he is capable of defending them:
"I say in truth that I have proved more might in the sea than any other man, and more endurance in the ocean. The two of us had talked in our youth and bragged -- we were still mere boys then -- that we would risk our lives far out at sea, and so we did it."
(Beowulf, eNotes eText)
For the time and the place, this sort of self-aggrandizement is completely acceptable, and in fact expected. For Beowulf to downplay his achievements would be seen as strange and falsely humble; if he is not willing to trumpet his abilities and his strength, he would not be seen as capable of fighting Grendel.