How did Britain govern the thirteen colonies?

By the seat of their pants.

Couple of points that are vital to remember.

1. As Jack Rakove points out, British colonial possessions were unique in that the colonists expected to settle and displace the natives. Other colonial powers had different philosophies. Since the colonists were permanent residents and owned real property (land), governance was required.

2. The British Empire (arguably) dates from the end of the Seven Years War. Even if you disagree, I would assert that the British Empire changes in nature at that point - Britain was catapulted to a superpower with effective dominance over the seas and extensive overseas holding. They hadn't planned for this and their government was certainly not designed for it.

3. The British government was still in evolution; not all the implications of the Glorious Revolution had been worked out. For example, the concept of the Loyal Opposition hadn't been formed yet, and they simply didn't have the concepts to understand and work with multiple groups of people who agreed on the ends, but vehemently disagreed on the means.

4. Everyone agreed that Parliament was supreme, but they didn't know what that really meant. Parliament was clearly incapable of exercising executive function, and the horde of bureaucrats (placemen) remained the province of the Crown. (Aside: One of the great mysteries is why the Crown failed to effectively exercise the placemen system - for example, there were very few Church of England appointments to the colonies)

5. Landholders in Britain were represented in Parliament, but there was no mechanism to represent landholders in the colonies. The rotten borough system inhibited any attempt to change representation.

6. The effect of the prior two points is that the de jure supreme branch of government didn't know how to exercise power in the colonies (There secretary of state for the colonies was a powerless position with no bureaucrats, little or no staff, etc.) There were no precedents (remember that precedent is vital in the British system), no bureaucrats, no way to tell what is happening and no mechanism to exercise policy. The branch of government that was inferior (the Crown) had bureaucrats, and had an excess of military officers (one of the policies was to settle half pay officers in the colony), but didn't really have a mandate. Neither branch had a plan or a goal for the colonies, except for mercantilistic exploitation (which we now know is a dumb plan)

So how did Britain govern the colonies? ad hoc, seat of the pants, incoherently, pick your favorite derogatory adjective. I think all parties would agree that they did it badly.

Unfortunately there were some serious stresses on the system.

1. The need to retire military officers to half pay. The Seven Years War was effectively the first world war, and the British Military grew to the task. Once the war was over, Britain had to shrink the military to control the budget. (remember that one of the key policy goals of Parliament was opposition to a standing military).

2. Treaties with Native American Powers. Britain had promised to prevent settlement in Native lands. The American colonists didn't just ignore that policy, they conspicuously flouted it. George Washington raffled off land across the Mississippi in a way that was probably treasonous. (Directly opposed to the policy of the King and Parliament). Colonists kept forming companies to exploit lands that were - by treaty with Britain - not colonial lands.

3. Much of the buildup to the Revolutionary war was formed around exactly how Parliament could govern colonies - could they levy taxes? Must they consult with local councils? (they didn't in Britain).

Britain governed the 13 colonies through trading companies that were much like the East India Company. The one for Virginia was called the Virginia Company. The one for Massachusetts was the Massachusetts Bay Company.

In order to secure the cooperation of the colonists, the trading companies often granted them unusual powers. For instance, the Virginia Company allowed Virginians to have a legislative assembly called the House of Burgesses in 1619. Likewise, the Massachusetts Bay Company also allowed the "Puritans" a great deal of self rule, including the right to choose their own leaders.

Sometimes the King would get concerned, and try to take back liberties that were granted to the colonists. For instance, in Virginia, executive officers (governors, county commissioners, sheriffs, etc.) were appointed by the King, through the Virginia Company. The House of Burgesses, however, made most (local) laws; except for rules governing the prices and practices of tobacco exports, Virginia's largest cash crop. Notably absent, until the 1760s, were attempts by Britain to tax the colonists in America.

Colonial liberties would wax and wane based on England's internal politics. The rise of Oliver Cromwell in England led to a relaxation of royal rules, as did the Glorious Revolution. On the other hand, the rules tightened after the "third" generation King George inherited the throne. He used the burden of defending the colonies during the French and Indian War as an excuse, as did a predecessor, King James II vis-a-vis Massachusetts, after King Philip's War.