There are a good number of reasons why the British were able to do so, and in fact rule over India effectively for over a century.
Disunity among Indian princely states. India was more a collection of warring princely states, at loggerheads with each other. The British sucessfully used this to play off one state against another. Add to it there was no dearth of people willing to betray the kingdom for a few pieces of silver. Robert Clive succeeded at Plassey, because Mir Jaffar was willing to betray his master Siraj-Ud-Daulah in lie of being the Nawab. Mir Jaffar himself was betrayed by Mir Qasim later on.Tipu Sultan one of the most redoubtable fighters against the British rule, was finally defeated, as the Marathas, King of Mysore, Nizam of Hyderabad all joined hands with the British.
Superiority over other colonial powers. The other colonial powers in India competing for the share of resources were France,Portugal,Denmark, Holland. Of the 4, Denmark and Holland could never really be serious competitors to the British, they had their own trading posts, scattered around, but were never a serious threat. Portugal focussed primarily on the Western coast, Goa, parts of Kerala, Karnataka, and this left the British with vast swathes of unoccupied territory. That left France as the major contender to Britian in the race for colonialism. The British Army was more well equipped, more professional, more disciplined compared to the French army, suffering from indiscipline and corruption. This made the British win key battles all over the East Coast, as they effectively grabbed control.
Doctrine of Lapse. One of the most effective tactics, the British used to take over most of India. Instead of waging an all out war against some of the princely states, they signed a treaty with them, where in if the ruling king died without a heir, the East India company could take over that. And that is how Satara became one of the first states to end up under British rule. And that was also the main reason for the conflict in Jhansi.
Subsidiary alliance was also an effective instrument. According to this alliance, the kingdom which signs the treaty will have to maintain the following rules:
The British agreed to maintain a permanent and fixed subsidiary force within the territory of their ally.
In return, they didn't take money but took over a part of the territory of the ally.
A British officer called "resident" was placed at the court of the ruler.{he could interfere in the internal matters of the kingdom}
The ally could not maintain any relation with any other ruler without the approval of the British.{so,when the rulers wanted to revolt against the British they are alone.}
The Indian rulers felt a false sense of security but in reality they were losing their independence. On the other hand the Britishers maintained large forces at the expenses of the Indian rulers and also increase their area of influence. Some states brought under control through this policy are Hyderabad, Tanjore, Awadh, etc.
At the end of it all, the British had the advantage of better manpower, were militarily more powerful and stronger, and add to it they had some very canny strategists too. And the disunity among Indian princely states, their constant warring with each other, just added to the advantage.
It helped to spread Enlightenment ideas.
