Comment by accurrent

A lot of things. The British empire is a master class in financial engineering (largely by luck imo). They started by raising capital via IPOs to create trading companies. The first successful ones were the ones that established the 13 colonies that became the USA and Jamaica.

While they maintained control over those, they engaged in war and looting against the established Spanish empire. They managed to entice the Indian elite by being extremely effective at paying back loans using the loot they got during the Spanish era. Internal division among elite in India meant that the bankers in bengal thought they could use the British for deposing a king they did not like and at the same time handle the raids that the Marathas were performing. The British essentially used the money to bribe the king's army and took control of Bengal (which was the ricebasket for one of the largest empires of the time and the largest cloth producer). Unfortunately for said bankers, once the British took control of bengal they betrayed the bankers who lent them money. They raised an army of local sepoys and went on to take the rest of the subcontinent (although this was a slow painful process).

Even as the US revolted and gained independence, the British found themselves with a growing colony in India where civil wars and divisions between kings made it easy for the British to move in. One thing to remember is that when we talk of a British army at this point it really is an army where the lower ranks are made of Indians who were trying to make a living (as it turns out British were just better paymasters than their local counterparts).

Once they conquered India, they moved east and use Indian soldiers to take on Chinese while waging the opium war (and making a tonne of money from the trade of drugs).

Ive glossed over a bunch of things but honestly the success of the British in the 1800s is built over many many years of multiple investments some of which started in the 1500s (for instance their acquisition of Mumbai was entirely by sheer dumb luck as it was part of a dowry given by the Portuguese, who in turn received it as part of a peace treaty with a local ruler)... Many of the things that brought them success was a result of fairly immoral behaviour. And while their governing class in Europe is fairly small they had massive local administrations run by locals reporting to them.