The british empire was a vast empire itself but during those days where kings and queens ruled the places..they wanted to expand their teriitory and establish their power over the whole of earth :) And they wanted to expand their power too thats why they came to india...
India was a prosperous nation by itself but it was uneducated and the british saw this chance to take over India as they thought indians to be backward ans incapable of governing themselves!!
Indians had a lot of precious stones and spices and cheap labour was also available..so british took advantage and invaded India to loot it
On top of it the mughals were already ruling..this clinched the dispute..
the distance between muslims and hindus was also gradually increasing and ultimately ..British overthrough the nation and established their power
By doing so they can also get their work done..for example they got their resources frm india during world war ,they could recruit indians fr war fighting,they also forced indians to plant tobacco and indigo
Before britishers invaded ..No indian was adicted to smoking and stuff