How did England acquire colonies in America?

England acquired colonies, in theory, by sending out explorers who mapped and claimed vast tracts of land in North America for England. Of course, simply mapping and claiming territory does not give a country much hold on it. England's wise move was to spend the money and take the risks of establishing permanent colonies on the North American seaboard. Some of these colonies, like Roanoke, failed. Some, like Jamestown and Plymouth, began as tiny toeholds. Once England had established several bases of permanent settlement, it became easier and easier to send more people to spread out.
Of course, Native Americans were here before the English, but contact with diseases such as smallpox for which they had no immunity quickly reduced their numbers, as did warfare. England entered into a number of treaties with various tribes to try to contain violence.
Another way England gained colonies on the eastern seaboard, at least in the case of New Amsterdam (which became New York), was simply to sail their fleet into the harbor and state that Francis Drake had claimed the land for England before the Dutch arrived, so it belonged to them. As the Dutch had no military defenses to speak of and had problems in Europe, they ceded the land to England.
Colonies became the spoils of European wars, parts of settlements to end fighting there. It was an easy trade for European countries to give away land far away in order to gain peace in their homelands.
In sum, England's growing military might, especially its powerful navy, helped it gain colonies, as did its willingness to send people to do the hard slog of actually settling in a hostile place.