Am astonished that the obvious answer hasn’t been uttered yet.
There was a longstanding practice in America of referring to white ethnic minorities as hyphenated Americans. (Something Teddy Roosevelt famously decried.) There were, and are, French–Americans, Italian–Americans, German–Americans, etc etc.
Blacks changed their description from racial terms to a hyphenated American term in the 1980s, to emphasise that their deviation as a minority from any American norm should not be treated any different from that of French–Americans: that they too were just another ethnic minority within America—but still an integral part of the national fabric. That they were an ethnic minority, not a “different race”, with all the exoticisation and 3/5 of a person baggage that entailed.
Black people in America are called African–American in order to assert that they are no different from French–Americans. The discrepancy in prevalence with the white hyphenated terms is one of timing: by the 1980s, those white ethnicities had largely assimilated, so their hyphenated terms weren’t as widely heard as they were in Teddy Roosevelt’s time. And the steep rise in the 1980s is unsurprising: it was a deliberate coinage at a particular time.
In the 1980s, the term African American was advanced on the model of, for example, German-American or Irish-American to give descendants of American slaves and other American blacks who lived through the slavery era a heritage and a cultural base. The term was popularized in black communities around the country via word of mouth and ultimately received mainstream use after Jesse Jackson publicly used the term in front of a national audience in 1988. Subsequently, major media outlets adopted its use.
Many African Americans have expressed a preference for the term African American because it was formed in the same way as the terms for the many other ethnic groups currently living in the nation. Some argued further that, because of the historical circumstances surrounding the capture, enslavement and systematic attempts to de-Africanize blacks in the United States under chattel slavery, most African Americans are unable to trace their ancestry to a specific African nation; hence, the entire continent serves as a geographic marker.