boast (n.)
mid-13c., "arrogance, presumption, pride, vanity;" c.1300, "a brag, boastful speech," from Anglo-French bost "ostentation," probably via Scandinavian (compare Norwegian baus "proud, bold, daring"), from Proto-Germanic *bausia "to blow up, puff up, swell" (cognates: Middle High German bus "swelling," dialectal German baustern "to swell;" Middle Dutch bose, Dutch boos "evil, wicked, angry," Old High German bosi "worthless, slanderous," German böse "evil, bad, angry"), from PIE *bhou-, variant of root *beu-, *bheu- "to grow, swell" (see bull (n.2)).

The notion apparently is of being "puffed up" with pride; compare Old English belgan "to become angry, offend, provoke," belg "anger, arrogance," from the same root as bellows and belly (n.). Related: Boasted; boasting. An Old English word for "boasting" was micelsprecende, "big talk."
boast (v.)
early 14c., "to brag, speak arrogantly;" from the same source as boast (n.). Related: Boasted; boasting.