English borrowed avant-garde twice. The first time it was a word for soldiers. The second time it was a word for painters.
In French, the avant-garde is the advance guard: the part of an army that moves ahead of the main body to scout and protect it. Avant means "before," from Late Latin abante, "from in front of." Garde comes from Old French garder, "to keep" or "protect." English used the term in this literal military sense from the 15th to the 18th century.
English also took the same French phrase in a worn-down form. Etymonline traces vanguard to an Anglo-French variant of avant-garde, recorded in the mid-15th century.
The leap from war to art came out of early French socialism. Henri de Saint-Simon believed artists, scientists and industrialists would together lead a new society. A text from his circle, The Artist, the Scientist and the Industrialist, cast artists as the advance guard. The Tate quotes its reasoning: "when we want to spread new ideas we inscribe them on marble or canvas."
Sources disagree about who wrote that line. The Tate credits Saint-Simon himself, in 1825. EBSCO and Wikipedia credit the 1825 essay to Olinde Rodrigues, one of his followers, and call it the first use of the term in this sense.
The word kept its political edge for decades. EBSCO describes the essay as arguing that art could reform politics, economics and society. The Tate dates avant-garde art itself to the 1850s and the realism of Gustave Courbet, who was strongly influenced by early socialist ideas.
English borrowed the word again in 1910, now meaning the pioneers or innovators of any period. It was working as an adjective by 1925. In 1939 the critic Clement Greenberg set it against mass culture in his essay "Avant-Garde and Kitsch." Today the word is close to a synonym for experimental art, though it still carries its original picture of a small unit marching out in front.