Why did the British come to the American colonies?

Firstly, they were not "American Colonies" at the time. They were bush and wilderness. Nobody laid claim to the land.
The first settlers were religious fundamentalists, who left Great Britain to be able to practice their own oddball form of religion, called Puritanism. They could be compared to the current crop of religious right wingers in the USA. They had little practical knowledge of how to survive in North America, and if it were not for the Aboriginals, they would have died of starvation during the first winter. What is now called " Thanksgiving" should really be called " we were really lucky to survive the first winter here".
The Puritans made a number of huge errors as they tried to re-create England in North America, in both the way they tried to farm the land and how they built their homes. They were too stubborn to adopt the proven methods of the Aboriginals when it came to living here. It is a wonder they survived at all.
Later they came to make money from plantations.
EXPANDED ANSWER: In 1587 the first settlers came to Roanoke Island in what is now North Carolina. They were not religious fundamentalists. The colony failed and the survivors left the island. It is not known where they went or what happened to them although there is a strong case that at least some of them went to the Chesapeake Bay area (in Virginia, where they were supposed to go in the first place) and were later massacred by Powhatan. There is also some speculation that at least seven escaped and were later enslaved by a local tribal group.
FULL ANSWER: Colonists came to North America for a variety of reasons. The most common answer given would be for religious reasons, and this was the case for many non-Anglican religious groups who were persecuted in England - these groups included the Puritans of New England, the Quakers of Pennsylvania, and the Catholics of Maryland. However, they also came over seeking wealth, as was the case in Virginia and the Carolinas, where tobacco, sugar cane, cotton, and other cash crops were grown and sold to England. There were also a variety of other natural resources to be had in North America, not the least of which was timber and fur, both of which were difficult to find in England. Debtors that had been placed into prisons because of their debt were sent to Georgia, where they also made a convenient blockade between the Spaniards in Florida and England's wealthy plantation communities to the north.