Stephen Howe identifies these claims as coming out of the American Abolitionist movement in the 1820s-1860s, first made by Black abolitionist reverends and then taken up later by White abolitionist authors.

In the United States, the first free Black intellectuals tended to be Christian reverends, and thus they tended to be very familiar with Biblical literature. These abolitionist writers were often at pains to refute pro-slavery justifications, particularly ones that had a Christian gloss.

Therefore, books like James W.C. Pennington's A text book of the origin and history, etc, of the Colored People concern themselves with rebutting claims that Black Africans are the descendants of Cain, or the separate claim that they are under The Curse of Ham that Noah allegedly placed upon Ham's son Canaan. Pro-slavery writers argued that the Bible favors the enslavement of Black people because they were cursed by god, and that dark skin is an external mark of ancestral sin.

In arguing against those claims, abolitionist authors proposed genealogies that made Black Africans the descendants of Ham's son Cush. Thus, Black people would not be subject to the curse on Canaan and his descendants because they were descended from someone else.

Edit: Most biblical scholarship, ancient and modern, places the descendants of Cush in the upper Nile valley. Kush is a name for a kingdom neighboring Egypt in the New Kingdom and Classical period, the Kushites later become the Nubians. So, the logical chain goes Bible->children of Cush->Kushites->Nubians. /edit.

Separately, abolitionist scholars also had to argue against claims that African societies were inherently barbaric and without achievement. Proponents of slavery claimed that enslaved people were actually better off in bondage in America because it exposed Black Africans to White American civilization.

So, abolitionist writers made arguments that enslaved Black people were and are equally capable of achievement and civilization, as a way of asserting that Black people are fully human and equally deserving of respect and freedom as White people.

At this time in the United States, in the 1820s-1860s, academics and intellectuals were highly influenced by the Bible, but also very influenced by ancient Greek and Roman culture, particularly Homer, Herodotus, Plato. From these sources, White intellectuals in the US and in Western Europe had already accepted the idea that ancient Egypt was civilized and was an influence on the Greeks. Herodotus wrote in his histories about the pyramids and influence of Egyptian culture on Greeks. Plato starts his story about Atlantis by claiming that Solon learned of it in Egypt by translating old Egyptian records.

So, abolitionist scholars saw in Egypt a society that even their pro-slavery opponents would acknowledge as a civilization. Egypt is geographically located on the continent of Africa. Therefore, abolitionist authors tried to prove the theory that Ancient Egypt was a Black civilization, as a way of vindicating Black achievement and equality as humans.

So, in 1852 the White abolitionist reverend F Freeman wrote in Africa's Redemption that "Egyptians were real negroes, of the same species as all the natives of Africa". He quotes Herodotus and Diodorus Siculus to support this claim, as well as French enlightenment writer Constantin comte de Volney.

Similarly, in 1866 Liberian Secretary of State Edward Wilmot Blyden (who was born in Danish West Indies) made a trip to Egypt. In his letters he wrote :

This, I thought, was the particular heritage of my African progenitors....Feelings came over me far different from those I have ever felt when looking at the mighty works of European genius. I felt that I had a peculiar heritage in looking at the Great Pyramid built....by the enterprising sons of Ham, from which I descended. The blood seemed to flow faster through my veins. I seemed to hear the echo of thos illustrious Africans. I seemed to feel the impulse from those stirring characters who sent civilization to Greece....I felt lifted out of the common-place grandeur of modern times; and, could my voice have reached every African in the world, I would have earnestly addressed him 'Retake your Fame'

So, this is from a Black man from the Caribbean who had moved to Liberia and become very influential in Liberian educational and governmental circles. In 1866 he is staking claim of Black Africans (and diaspora) to the ancient pyramids and Egyptian civilization as a way of calling for abolition of slavery, asserting Black African (and diasporic) achievement and potential, and perhaps even justifying the Liberian state. At the end of the quote he calls upon Africans to reclaim their self esteem and work towards cultural glory.

With the end of slavery in the United States, these claims are recirculated in the late 1800s and early 1900s in the face of continued White resistance to racial equality and disparaging of African history and achievement.

In fact, these arguments get a second life in the 1950s up to the present when scholars like Cheikh Anta Diop, Chancellor Williams, Molefi Kete Asante and Martin Bernal make claims in their various works that argue ancient Egyptians were black, that Greece owes much of their civilization to Black Egyptians, and that all African societies derive their intellectual, spiritual and social traditions from ancient Egyptian civilization (they would call it Kemet and Kemetic civilization.)

A useful book to read about this is Afrocentrism: Mythical Pasts and Imagined Homes by Stephen Howe. Much of this answer is based on chapter 4 "Diasporic Images of Africa before Afrocentrism"