New rulers intentionally kept the borders to "avoid disruptive conflict amongst themselves". There was also a strong Pan-African movement amongst Africa's independence leaders, and many sought to consolidate centralized rule over the diverse new nations that had been forged, rather than fracturing them into multiple states.

After independence, there was a large focus on creating new national identities at the time, which is a very interesting subject in itself. African leaders attempted to build a unifying national identity to surpass all tribal and ethnic divisions, with varying levels of success. This is evident even in the names of the countries: Ghana, Benin, and Mali are all named for historical empires that had united large swathes of land. Tanzania merged its colonial names Tanganyika and Zanzibar to form a new name and to unite the people under a new identity.

Additionally, many African nations adopted a single-party structure or dictatorship in the post-colonial period, because they knew that a multi-party system would end up dividing the country along ethnic lines and would cause the state to weaken and disintegrate into ethnic conflict. However this has ultimately happened anyway in many countries once their dictators died or were deposed.

So essentially, it was due to the belief in national unity. Africa's post-colonial leaders sought strength in unity to overcome ethnic conflicts which had been deepened by the "divide and conquer" European colonialist tactics. They hoped to consolidate their control over the country's resources and economy to develop Africa and make their countries economically and politically strong.

Source:

"Africa in World History" by Erik Gilbert and Jonathan T. Reynolds

In broad theoretical scope: Nationalist African political parties generally won independence by capturing the 'state' (the institutions of governance). These institutions were established and operated to manage a territory -- infrastructure, political divisions, constabulary districts, etc. By capturing the state, victorious nationalists, in effect, became captured by it. Hence the boundaries of the state remained.

To put the responses already given into perspective, even Hispanic America (once independent from Spain in the 1820s) maintained largely the same borders. They followed the principal of "uti possidetis juris"; basically, what you possessed before independence you had a right to afterwards.

There were exceptions, but these were looked upon very critically as giving negative examples of fragmentation which other territories might try to emulate. Bolivia had to name itself after Bolívar and declare him perpetual president of the nation whensoever he stepped on Bolivian soil in order to butter him up enough to accept it breaking away from Peru.

The fear was that failing to follow the arbitrary borders already traced during the colonial era would lead to fragmentation (as occurred to Central America) or to conflict among neighboring states (as occurred between Chile, Peru, and Bolivia).

Would trying to change the borders have actually reduced ethnic strife, or just created an opportunity for small conflicts to spiral into ever-larger wars?