"The term in its present form is 19th century. Anthony Trollope, in his work Australia and new Zealand, 1873 observed:
'In the colonies... when a gentleman sounds his own trumpet he 'blows.'
That is clearly commenting on what must already have been a well-known phrase, although there aren't known printed citations to support that from before 1873." Read more about it here.
When people use the phrase they are saying the person is being boastful and self-promoting.