How does the French and English colonization of African civilizations demonstrate the idea that history is written by the people in power?

Well, I think we should start off by taking a step back and first asking: Does this colonial history demonstrate the idea of history written by the people in power?

My (unsatisfying) answer is both yes and no.

In West Africa, or really anywhere that had an experience with Islam, there already existed written historical chronicles like the Tarikh al-Fattash, Tarikh al-Sudan, the Kano Chronicle. Along the Swahili coast there was similarly the Kilwa Chronicle. And we also have surviving works from medieval scholars and travellers like al-Umari, Ibn Battuta and Ibn khaldun.

So, continuing to focus on West Africa, scholars like Maurice Delafosse were not writing a new history out of whole cloth, but instead he and Octave Houdas translated the Tarikh al-Fattash from Arabic to French and included their own commentaries on the text.

To be sure, the Tarikh al-Fattash was probably written by Ibn al-Mukhtar (or else his grandfather Mahmud Kati). In either case, those men were important, powerful figures in the later Songhai empire and early Moroccan period in the 17th century. So, you can still argue that work (and Tarikh al-Sudan and Kano Chronicle) was written by people in power. But, at the same time, in the Colonial moment, French scholars were guided by existing pre-colonial written works rather than creating new histories from scratch.

Also, aside from these written accounts there were additionally the oral traditions passed down by griots, who were professional musicians and keepers of lore in West Africa. The Epic of Sundiata is one example of this oral history which was collected and set to print in this colonial era.

But, we have to remember that simply printing the Epic of Sundiata did not give European scholars a monopoly on historical memory. Griots continued (and still continue) to have a function as keepers of memory, and could tell stories that contradicted or complicated "official" French histories. Since Griots could communicate in local languages like Fula, Bamana, Soninke, Mandinka, their songs often had greater popular influence among non-literate peasants than printed accounts in French.

This also brings me to another aspect of the question.

Who is the audience for history? Is it only history if it is written?

To be sure, the French and English colonial scholarship produced a historiography that laid the groundwork for African Studies or African Histories as disciplines in Western universities.

Important foundational figures to African Studies in the United States and UK, like Jan Vansina and Terrence Ranger, started their careers in the late 1950s and so benefited (maybe unwillingly) from privileged places in the Colonial hierarchy during their early fieldwork.

On the other hand, again, academic history is not the only history. Africans had and have their own historical memory, which could be expressed in text or through oral traditions or even rumor. For instance, take Luise White's book Speaking with Vampires which deals with colonial era rumors that fire departments in Tanzania, Uganda, and Kenya acted as "blood collectors" for European vampires.

Or, A Dance of Assassins by Alen Roberts does a great job of examining historical memory of Major Emil Storms and his time in the Congo Free State. In the book, Roberts first explores Emil Storms own diaries and reports, and contemporary Belgian, English and New Zealand newspaper accounts of Storms exploits.

Then, Roberts incorporates the interviews he conducted with Tabwa elders Luis Mulilo and Kizumina Kabulo concerning what they remembered and what rumors or traditions had been passed down about the conflict between Emil Storms and the Tabwa warlord Lusinga lwa N'gombe.

So, this is another case where Belgian written accounts told one story, and that would produce a history accessible to Belgians or Britons or Americans. But at the same time, the Tabwa people and their neighbors along Lake Tanganyika have their own memories of Storms and Lusinga, producing a different history for a different audience. And Belgian colonial rule could not impose a different, single historical understanding of those events in the 1890s.