It is inaccurate to say that the British Empire fell, rather the nature of its global influence changed relatively slowly over a span of decades. This happened for two reasons: the development of the Commonwealth (in contrast to the Empire), and the rise to prominence of other global powers, including and especially the US, USSR, and China.

The idea of the Commonwealth as opposed to Empire was both an internal political movement in British parliament, first observed by Lord Rosebery in the late 19th century and growing into its legitimacy by 1917, and among the colonial dominions of the empire themselves. The premise was put forward in response to Rosebery's observations that many of Britain's colonial territories - Canada and Newfoundland, Australia, New Zealand in particular.

By the time WWI rolls around, all of these states were functionally independent of the Empire, but the political ties were still extremely close (much is made in Canadian history courses of the assertion of independence Canada made by waiting a whole day before joining Britain in declaring for WWI, to say nothing of the whole week we waited before WWII). This ties into the development of global politics at this point. The New World was no longer the new world. The Middle East and Africa were the New New World and the focus of global exploration and development. Britain was fostering colonies here - Somaliland, Rhodesia, British Palestine, etc, but there was change in the wind about how protectorates like this were viewed, and the idea was essentially to let these little birds spread their wings and fly.

By post-WWII, formal independence was becoming a reality for the Big 3 (Canada, Australia, New Zealand) and in the places where it wasn't happening through diplomacy, it was happening by force. Ireland was on the fast track to forcing its independence from Britain, and Gandhi was reinventing the Imperial relationship in the New New World by essentially forcing meetings with the British monarch about independence for India. The way the world worked was changing in big ways.

As for the emergence of new powers, I'm sure I don't need to explain this in too much detail. The national militarizations, economic windfalls, and political leverage that countries like the States, the USSR, and China gained from the World Wars (especially II, particularly for those latter two) sent them heading up while Britain was slowly making its way down.

As they emerged as the new global superpowers, international priorities shifted. There wasn't really a New New World anymore, no new places where an uncontested sphere of influence could be established as the European powers had been trying to do in Africa and the Middle East, and America, Russia, and China probably wouldn't have been particularly interested in it even if their was. Russia wanted to see the influence of Communism spread, the US wanted to see it contained, and China didn't necessarily want either so much as it wanted to be the dominant power in the Asian sphere of influence.

As it so happens, the Asian sphere of influence became the focus of those three powers, with the US competing with Russia and China for allies (Korea and Vietnam are the most obvious examples of this). Britain's most powerful one-time colonial holdings had achieved a political independence and in Canada's case were more closely tied with America than Britain by then, and as contested territory shifted east, their holdings in Africa and the Middle East became increasingly less valuable in terms of their global political leverage.

The idea of the Commonwealth endured in one form or another into the 1970s, but as post-colonial theory became more prominent and the African and Middle Eastern holdings began looking for independence without ever becoming major players on the international state, it was becoming more costly for Britain to try and hang on to them than it was to let them go it alone.

TL;DR - Britain's most powerful colonial holdings began asking for/demanding their independence, those that remained were eclipsed in global political value by the neo-colonial holdings of the post-war superpowers.

I admit this post is done mostly from memory. Let me know if you want sources and put a short bibliography together.

The British Empire can be roughly divided into two distinct parts: the Dominions (British settler communities in Canada, Australia, New Zealand and South Africa) and the rest (mostly tropical territories in Africa and Asia).

Following the Canadian Rebellion the British felt they had learnt lessons from the loss of the American colonies by granting Canada self-government within the Empire, as embodied in the famous Durham Report. The rest of the British settler communities eventually followed suit so that by the twentieth century, the relationship between the Dominions and the UK was already one of semi-detachment (they erected tariffs against British goods, for example). Proposals for imperial union or federation floundered on Canadian and South African hostility to such schemes.

Due to Britain's industrial decline she was unable to manufacture or purchase the necessary armaments for herself or for the protection of the Dominions during WWII. The Dominions now looked to America rather than Britain for protection as Britain was now too poor to do so. For example, in 1941 the Australian Prime Minister said publicly:

The Australian Government, therefore, regards the Pacific struggle as primarily one in which the United States and Australia must have the fullest say in the direction of the democracies' fighting plan. Without any inhibitions of any kind, I make it quite clear that Australia looks to America, free of any pangs as to our traditional links or kinship with the United Kingdom.

India was granted independence after the War by a bankrupt Britain governed by a Labour Party that was instinctively anti-imperialist. For the rest of the Empire, this was mostly granted independence in the late 1950s/early 60s. Harold Macmillan, the British Prime Minister, conducted an audit of Empire in 1957 in the aftermath of Suez and found that most of the territories Britain held were unprofitable. Therefore he concluded it would not damage the British economy if these were disposed of.

The two main causes of the disintegration of the Empire are the sentiments of the rulers of the Dominions who opposed imperial federation (which would have united the UK and the Dominions into one state) and the economic decline of Britain. This meant that Britain was unable to act independently nor protect the Dominions in wartime.