How did Britain benefit from America?

The American colonies were a benefit because they added to Great Britain's wealth and prestige.
Even in the seventeenth century, when Europe was far less populated than today, it had depleted many of its natural resources. North America, on the other hand, looked to the Europeans like a vast "wilderness" and was, compared to Europe, filled with an abundance of seemingly limitless natural resources: fertile land to be cultivated, a vast number of animals for fur, trees for timber, and other raw materials. The merchant and governing classes saw the potential for vast money making.
The British set up the colonies to capture most of the wealth produced for the British homeland, which eventually created the tensions that led to the American Revolution. Legally, the colonists were only allowed to sell their raw materials to the British. Since it was a captive market, the British could pay low prices for these materials. (There was such an abundance, however, that the colonists could still prosper.) The British would then turn these raw materials into finished products and sell them back to the Americans at high prices. The British had a monopoly on selling as well as buying, as the colonists were, in theory, only allowed to buy from the British."Buy low, sell high" was the motto.
If they did buy finished goods from other countries, the colonists were supposed to pay high tariffs. For many years, however, the British turned a blind eye to the Americans breaking these laws, as there was so much wealth pouring in; when the motherland did try to get more serious about enforcement of tariffs, the Americans became extremely unhappy.
As for prestige, after realizing that Columbus had discovered a new continent, the race was on to establish first outposts, claims to land, and then colonies there, which added to the prestige of countries wanting to be powerful players on the world stage. This period of colonialism and imperialism would last into the twentieth century, after which the cost of administering colonies began to outstrip profit. England, a rising world and a naval power, wanted to flex its muscles around the world and show it was a power to contend with that could control the high seas. Colonies in the Americas helped it extend its power and prestige.