There were economic realities and political movements that contributed to African nations throwing off their colonial masters as well.
Economically, the two largest empires in Africa by the post-World War II era, France and Britain, went broke. England was mostly broke after World War II, and it was no longer feasible for them to project military power around the globe, or fight wars of subjugation on the African continent. France had suffered economic devastation under Nazi occupation, then fought a protracted and ultimately losing effort in Indochina, and then an insurgency in Algeria. Their empire collapsed as a result, and the influence of both nations was replaced by that of the US and the USSR.
Politically, Pan-Africanism, a movement to unite African peoples and erase colonial borders, became more popular in the 1950s and 60s, as did Marxism, which many African supported as a means to address the glaring economic differences between wealthy colonial masters and the native peoples of that land.
So it makes sense that 1960 was considered the "Year of African Independence", as seventeen separate nations declared their independence.
In general, Africans were successful in gaining independence from colonial rule because the time had come. The time when world opinion would accept colonization had gone by the 1960s and so the African countries were able to gain their freedom.
The major changes happened with WWII and the Cold War. After WWII, the United States became the superpower of the Western world. Since the US did not approve of colonization, it put pressure on its allies to free their colonies.
The Cold War moved this process along. The West's Cold War rhetoric about freedom and democracy demanded that Western countries free their colonies. The West could not talk about the moral superiority of democracy and freedom while still keeping colonies.
Because of these things, the 1960s were a time when it was less and less possible for European countries to keep African colonies. This led them to give their colonies up.