Mostly it comes down to a confluence of three factors that came together to give the EIC the chance to mostly subsume India, the collapse of the Mughal Empire, a window of strong military superiority, and luck.

The collapse of the Mughal empire that had ruled most of India for quite a while. For a long time the Mughals had ruled by being relatively tolerant of the Hindu majority despite being Muslems themselves. This started to fall apart towards the end of the empire and when combined with military over-extension caused rebellions to break out from 1707 onwards. Civil strife escalated and by 1719 you had a year where 4 different emperors sat on the throne. Many governors remained nominally loyal to the Mughal court but had stopped payment of tribute and in effect the Mughal emperor ruled only the area around Dehli.

The East India company had been founded in 1600 to trade in the Indian ocean. Military force had always been part of the equation, however, the first EIC expedition involved the capture of a Portuguese ship and its cargo.

The EIC was allowed to set up a trading station at Calcutta in Bengal in the northeast of India. Due to hostilities with the French the British started building up the fortifications of Calcutta which was against the terms under which they had been allowed to settle there and the Nawab of Bengal, Siraj ud-Daulah, decided to teach them a lesson. I won't go into the details but he was mostly defeated for two reasons.

The first was that at that moment Europeans had a number of important military innovations developed during the 30 years war such as volley fire, bayonets, close order drill, and mobile field artillery that gave their forces a substantial advantage. That hadn't been true when the Portuguese offended the Mughals a hundred years earlier and by 1800 or so local Indian forces were able to assimilate these ideas. But there was a window of opportunity where European forces could defeat Indian forces many times their size and the British East India Company was well positioned to take advantage of this.

The other was that Siraj ud-Daulah's was widely hated. Think Joffrey from Game of Thones and you won't be too far off. One very important group he offended was the Jaget Seth, the wealthiest banking family in India, when he struck a member in public. Quite possibly the wealthiest in the world, at this point India was incredibly wealthy especially the Bengal region and had more manufactured outputs than even China.

Through force and cooptation of his allies the East India Company was able to effectively control Bengal in 1757. Once in power they taxed ruthlessly, causing a huge famine later in 1770. The word "loot" also entered the English language at this point from the Hindi word "lut" meaning more or less the same thing. Using the money, their military advantage, and clever diplomacy they were able to absorb many other Indian states. The last big obstacle to their consolidation of power in India was the Tipu Sultan of Mysore but while he had an army that could take on the British in equal numbers the British had too many troops and too many allies and defeated him and established hegemony over India even if they didn't rule the whole thing directly.

Parliament didn't like the idea of a private company wielding so much power and had been working to bring the Company more under its control since the first Regulatory Act in 1773.

Eventually there was a large rebellion in India in 1857, Parliment decided they were tired of the EIC running things, and simply nationalized the company in 1858.

Sources:

The Anarchy by William Dalrymple

Warfare in the 18th Century by Jeremy Black

I dont consider myself an expert on the subject, but I am currently working on my masters dissertation involving the trade of the EIC and growth of the British Empire. My area of focus is the companies operations in India; but it is important to note that despite its name, the East India company operated from the India and Burma, right up through the East Indies and into China and Japan. My inability to provide a complete overview will hopefully do something to explain the magnitude of this enterprise and subsequent problems with studying it.

The East India Company was not the first or only joint stock company to be created in the 16th and 17th centuries; others included the Virgina company (responsible for founding Jamestown in the US) and the Royal African company. These were not limited to England either as the majority of the large European powers such as Spain, Portugal and France also had large Joint Stock Companies. The EIC was arguabley the greatest of the joint stock companies, and understanding these companies is vital to understanding the systems that allowed these enterprises to thrive. Firstly there needs to be made the distinction between the English East India Company (the EIC), and the Dutch East India company (the VOC).

Joint stock companies began to appear at the end of the 16th century as the period of mercantilism was on the rise but had not yet transformed into the capitalism as we know it today. The main differences between mercantilism and free trade capitalism is that under mercantilism there was a large amount of government controll, and large enterprises such as the EIC needed to acquire charters from their respective governments in order to trade globally. Joint stock companies became very popular in this period as they were the first form of what we could call capitalism to arise, as they allowed private individuals like you or I to invest in ventures and trading missions and subesquentally make a profit should the venture be successful. This was a revolutionary way of sourcing finances at the time as it meant that the risk of a failed enterprise was spread out over hundreds of people, rather than one single investor who would have lost everything. This new process not only meant captains, merchants and explorers no longer had to search for prominent aristocrats or the crown for finance, it meant that they had access to essentially unlimited wealth provided more investors could always be found. The new joint stock companies brought together ships, investors, traders and workers to form a network that spread the risk across thousands of individuals, limiting personal risk and increasing the net worth of the company with every investor, ship, or explorer added to its ranks. This limited risk and increased profitability of the trading ventures only caused more and more investment to pour in. This is what started what I would call the capitalist revolution; the processes of merchants and traders freeing themselves from the restrictions imposed on them by feudalism and monarchs, and led to Adam Smith's writings in 1776 that began the free trade capitalism that we experience today. It is this unlimited source of investment that caused the companies to balloon and grow to unimaginable sizes even by today's standards. Obviously with our modern day understanding of economics we all know that eventually uncontrolled and unlimited growth will lead to a recession, however mercantilism did not operate the same was capitalism and since there was no limit on the size of a market or enterprises ability to operate in in during this period of time, the doubt and hesitancy that leads to every major recession did not happen, and the companies continued to grow exponentially. It wasnt until after the age of discovery that the world markets started to fill up, and the aggressive expansion policies no longer worked; thus leading to Adam Smith's theories of the Wealth of Nations and the capitalist system needed to preserve them.

This doesn't however explain how the English East India Company grew to be the dominant global power. The growth of the EIC links in with British Imperialism as the company, despite being made up of free public investors, had obtained an official charter from Queen Elizabeth in 1600 to trade in the name of England, and this meant that despite it being a separate entity and company it represented Englands trading interests abroad. These British companies (EIC, Virginia company etc.) were tasked with venturing out across the seas to open up new markets for the British to compete with the other European powers, and subsequently these markets then had to be defended. The EIC in particular amassed a large mercenary army (mainly made up of local sepoy soldiers) to defend these markets and this eventually turned into total controll of the market and country. The EIC took control of India and ruled in the name of the British empire. It wasnt until the British government began to recognize the threats of large companies with private armies larger than your own that they decided to remove the EICs charter and instead implemented the British Raj as protectorate. The EIC lost its comercial monopoly in India in 1813 and from this period on was subjected to more and more British governmental controll to limit its power and restrain it. By the middle of the century due to a slow down in investment, the company began to experience financial issues and was officially dissolved in 1874.

As I said at the beginning I am not an expert in this field, and unfortunately there is no straight answer to your question. The EIC was the largest enterprise the world has ever seen but how it got that way is still up for debate. I intend to argue in my dissertation that trade was the main reason, and specifically the trade of drugs and narcotics such as tea, cannabis and opium. However other scholars will argue that the growth was fueled by a nationalistic view to dominate the globe above the european competition of Portugal and France, or as a result of the increasing thirst for knowledge and exploration following Sir Francis Drakes round the world voyage in 1577-80.

For a good overview of the East India company I would recommend The East India Company: A History by Philip Lawson (Longman Group UK, 1993). K.N. Chaudhuri has also published a study of the early years of the East India Company that explains its rise simply called The English East India Company (Frank Cass & co, 1965), and The Honerable Company by John Keay (Harper Collins, 1991) also provides an in depth history of the company.

I am sorry if parts of my answer are unclear, so please feel free to querie further anything I have said.