It is debatable whether or not African Americans began to believe the stereotypes about themselves that have been popularized among the white communities. The assertion that they did so can be seen as another form of stereotyping.
Over the years, one has heard prejudiced people say, not only about African Americans but about other minority groups, things to the effect that "They believe (or say) it themselves." Usually those making these assertions have had no specific contact with people in the targeted group and therefore no way of knowing if this is true.
In general, stereotypes are created and disseminated partly out of fear. There has been a tendency throughout history for humans to reflexively be afraid of those who are different from themselves, to consider themselves threatened by others whose language, religious beliefs, or skin color are not the same as their own. Holocaust survivor Primo Levi in his book If This is a Man wrote that "whenever a stranger is perceived as an enemy, it leads to the existence of the Lager [concentration camp]." In many ways, the settings in which enslaved people in America were forced to work were a kind of Lager in themselves.
Stereotypes are a partly unconscious way of rationalizing the mistreatment of those who are different. If one can convince others that a certain group exhibits behaviors and characteristics that are alien and undesirable in some way, the marginalization and/or oppression of those people can be seen as a form of protecting the dominant ones. Behind this generalized fear of the "stranger" is the fear that "mixing" with them will dilute the supposedly superior traits of the rulers and eventually wipe out the ruling group's identity. Racial thinking, then, is aimed at exaggerating the differences with a group who have been subordinated, and this perpetuates extreme and negative conceptions of the victims.
If the victims themselves do begin to believe the false or exaggerated notions about them, it's because their own culture and identity have been suppressed, creating a vacuum where the rulers' thinking is the principal thing available to them to believe in. As is well known, this was done to Africans when they were brought as enslaved people to America. But the fact that African Americans did sustain their hopes for freedom and equality, even amid those conditions, is a testament to the ability of the human spirit to overcome even the worst obstacles.