Many of the territories of the British Empire, notably India and Pakistan (which was then part of India) had Muslim populations. When the Empire finally collapsed, these countries were often thrown into chaos as different factions vied for power. Some Muslims who were, by virtue of the occupation, subjects of the British crown, sought sanctuary in the British Isles. Later, these were joined by economic migrants and both populations bred more rapidly than the native British, so by century's end they were a significant minority in the country.
British Muslims are in the news today because many of them are unassimilated. That means that though they enjoy British stability, they have failed to understand or obey the laws and morals of their new country. This results in their committing crimes and making demands on the welfare system that burden the non-Muslim citizens of Britain.
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The same reason as other groups - for a better life
Also, sometimes they come for arranged marriages
Also they come to do business, visit family and friend etc