I am speaking in an African Context here due to my area of study. I think what you're referring to is why decolonisation happened in the post-war period. I will tackle some of the issues with framing the question the way you have in my answer.

Its important to recognise from the start that not every European Colonial Power relinquished their colonies between 1940s-1960s and they certainly didn't all do it willingly. Perhaps the most poignant example is in the Portuguese colonies (modern-day Angola, Mozambique and Guinea-Bissau). These colonies were not independent until the mid 1970's and achieved this through revolution/wars of independence. As such Portugal is the outlier to your question, it was forced to give up its colonies through warfare and revolution and didn't do so until roughly 10 years after the majority of other colonial powers.

That said the Portuguese were not the only to colonial power to experience wars of independence on the African continent. Britain had several colonial rebellions including the relatively prolific 'Mau Mau' rebellion and France was embroiled in a prolonged guerilla war with Algeria. The Algerian war would eventually spread to the France itself, not a good outcome politically for the French. While some colonies (Such as Ghana) did achieve independence through peaceful means, many more used a mixture of politics and violence. To portray Decolonisation as peaceful process in the majority of cases is wrong. In many colonies, even if the violence wasn't there, the potential threats of violence were and with them possibility of this spilling home.

One of the often stated reasons that countries other than Portugal did not retain their colonies to the bitter end is their adherence to the ideology of liberal democracy. In the post war countries, such as France, Britain and Belgium, imperialism as an ideology became less popular and the right to African self determination became a much more popular. To add to this following the horrors of genocide in the Second World War, it wasn't really possible to kill thousands in the name of imperialism. There were certainly imperialists within colonial powers who wished to retain the empire, Winston Churchill being one of the most famous, but the majority of European leaders adhered to decolonisation. By contrast Portugal remained very authoritarian and retained its imperialistic beliefs. The Salazar regime was not fond of concessions.

It is also important not to remove the agency of Africans from the equation. Movements for independence became more popular, both those committed to political and violent overthrow of colonial regimes. This was augmented by the Pan-African movements and anti-racism, that revealed that Africans were not inferior to whites. The majority of colonial states had small European populations, so often a class of African administrators were used to aid in the running of the state. These administrators often became the first African politicians and future leaders. To the outside world Africans could no longer be misrepresented as tribal barbarians (a major part in enabling colonisation around 100 years prior).

In short decolonisation was a very complicated issue within Africa. Looking further afield I expect that you will find it becomes even more complicated. Changes in ideology within Europe and Africa and the rise of the African politician class certainly played a part. That said it certainly was not all willing on the part of Europeans, violence and conflict had a hand too. If you are interested in a more global look at decolonisation Decolonisation the British Experience since 1945 by Nicholas White is a great look at British decolonisation and relatively accessible.