Why Did The Britishers Came To India?

The first and most basic reason would be for economics.  Following--and during--the age of exploration, certain governments quickly realized the potential for economic growth through trade and colonization.  

Commerce was greatly important to the growth of the British Empire.  Goods from the East were highly valued and the laws of supply and demand, as we understand them, came to the fore.  Through having a supply of exotic goods, Britain was able to create a demand (and vice versa).

Coupled with the growth of Britain's economy was the growth of its military prowess, with which it could defend its economic interests and spread more of its colonies.  This is how, by the Great War, the British Empire covered one-fourth of the globe.