Well, let's start off by noting that there are very many historical African states/societies. As such, different states had different methods. What was typical in one era and one point in time might be atypical in another corner of the continent.
Now, speak to the substance of the issue.
There were several civilizations in Africa that did build monuments and art and buildings out of wood. Ancient Egypt, Meroitic Nubia, Aksum, Zimbabwe and their Butua successors, and the Swahili^1 cities are just some examples of this.
Then, if we look at the Sudanic region of West Africa, the early Tichitt culture is known to have built stone animal enclosures and dwellings from 2600bc-400bc that set the precedent for West African urbanism^2. However, building methods in the Middle Niger region shifted towards the use of mud adobe and wood. This was partly because the raw materials, earth and water, are very abundant. Partly again, adobe is a better insulator, and a house made of mud is more comfortable in a warm climate than stone.
Which is not to say that no stone buildings were built in the Western Sudan. There are remains of stone buildings at Kumbi Saleh and at Awadaghust, along the trans-Saharan trade routes. However, at the Timbuktu of Mansa Kankan Musa, or the Gao of Askia Muhammad, monumental architecture generally followed the local architectural style. See the mosque at djenne or the sankore mosque
There are also examples like the earliest settlements at Mbanza Kongo in what is now Angola, where Portuguese accounts tell of houses made of wood, mud and thatch, with raffia cloth tapestries inside the houses used as partitions and for decoration^3. Such light construction enabled houses to be built or demolished quite quickly, enabling households or even entire settlements to remove and re-establish elsewhere.
So, in this context where whole settlements could move, either when farmland was exhausted, or there was a pestilence, or at the death of a King; the construction of stone monuments to represent royal power did not seem a sound investment.
This changes for the Kongo Kingdom when the Portuguese arrive, and we see the transformation of Mbanza Kongo into a city with several stone churches and a stone wall establishing the city limits. This shift is partly driven by exposure to Portuguese ideas of kingship and city living.
1 The African Urban Past: Historical Perspectives on the Metropolis by Susanne Preston Blier
2 Oxford handbook of African Archaeology Chapter 55- "Complex Societies, Urbanism and Trade in the Western Sahel" by Kevin MacDonald
3- Africa's Urban Past chapter 4- "Mbanza Kongo" by John K Thornton.
