The expression “to blow one’s own horn” means: To advertise oneself; boast of one’s own abilities; brag.
In England, the same self-advertising is done by “blowing one’s own trumpet,” and there is every reason to assume that this saying, or variants thereof, was the source of the American phrase.
Fleming, in A Panoplie of Epistles (1576), “sounded” the trumpet of his own “merites,” and writers of the eighteenth century “blew” their trumpets.
The earliest American usage, according to Bartlett’s Dictionary of Americanisms (1877), was “blowin’ his bazoo,” which was defined, “gasconade; braggadocio”, terms meaning boastful talk.
At some time since that date, and on a guess we’d say about seventy-five years ago, the slang term “bazoo” began to drop out of favor and “horn” became the accepted substitute.