In answering this question, I will assume that the “beginning of the American Revolution” dates back to the Stamp Act of 1765. In other words, the “American Revolution” for these purposes, is not the same thing as the Revolutionary War. Instead, it includes the political conflicts between the British government and the colonists that led up to the war.
There are three main reasons why the British came to see the colonies as a potential source of revenue in the 1760s. The first of these reasons is that this was a time when the British government was deeply in debt. The British had, in 1763, defeated France in the Seven Years’ War. This war had been very costly and the government had had to borrow a great deal of money to pay for it. Therefore, they needed to find money to pay off the debts. This led them to think of the colonists. They thought of the colonists in part (this is the second reason) because they felt the colonists had benefited from the war. The war had allowed many Americans to make a lot of money supplying the British soldiers. It had also driven the French out of North America, opening more territory for the colonists. The British felt the colonists had benefited from the war and should therefore have to pay their share of the costs.
The final reason has to do with the relative tax burdens on people in the colonies and in Britain itself. Although the American colonists would complain about being taxed, they were actually taxed much more lightly than the people living in Britain itself. They also tended, on average, to be better-off than those in Britain. Therefore, the British government figured that the Americans could afford it and that they really should be paying more in taxes.
For these reasons, the British government looked to the colonists as a source of revenues.