Cxi ne estas malbona ideo, sed gxin havas iu limigoj. This is not a bad idea, but it has some limitations.
The good part is that Esperanto would be easier to learn. While English, French, and other European languages are official languages in many African countries, many people cannot speak them properly. In some ways the European languages, the colonial lingua francas, have remained the languages of the cosmopolite elite as they are generally the ones who have been afforded the opportunities to learn them. Moreover with the curricular changes in some countries to primary school instruction being in local languages primarily (say in Ghana and Tanzania), English and French will become even more imperiled as languages used fully among the masses.
This is not necessarily a bad thing. By that I mean that the decline of the old colonial languages, while dysfunctional in the sense that it removes a common medium of communication, also allows for that role to be supplanted by other languages. In some places this transition is taking place organically. Take the spread of Asante Twi in Ghana as it is quickly becoming the defacto lingua franca of that country. Swahili is also slowly spreading into central Africa and north and south beyond the bounds of Kenya and Tanzania. Again these are promising developments as English (or French) were insufficient as organic mediums of communication in many respects.
This question of organicity is in fact the challenge of Esperanto. Esperanto does not necessarily have an organic base from which it can naturally spread. There are some groups in Africa who are teaching the language. On Facebook recently I was intrigued to see an Esperanto-Swahili dictionary, but the language lacks a foothold. This is something that can only be mediated by institutions or a living and thriving speech community.
I think that Swahili is a stronger contender for an indigenous language that might become a continental language, and in some ways this would be better. While no one owns Esperanto, it is still a European-derived language, and its advance in Africa does little to satisfy the decades old challenge of “modernizing” African languages as tools of creative and intellectual production (read Ayi Kwei Armah’s “Our Language Problem”). The fact that creative writers, legislators, and scientists still rely on Western languages to capture and communicate their knowledge would not be solved by the adoption of Esperanto. They would simply move their dependence from one European language to another (albeit an artificial one), while failing to facilitate this necessary development in their indigenous languages. I would consider such an outcome to be a profound waste of potential.