The power of the purse refers to the ability of the colonial elected legislatures to vote on public monetary issues in the colonies. At the time, the governors of the various colonies held a great deal of control in colonial government. They had the power to convene or dismiss the legislature and also to approve or veto any new bills. The power of the purse gave colonial legislatures the ability to have some control over government spending and the salaries of certain public officials, often including the governor. When the British took this power away from the colonists, it increased tensions, as the colonists had lost their check over the governor. As tensions rose and colonists felt they had lost significant influence in their local governments, calls for independence rose.

During the colonial period, American colonies were typically ruled by governors who were appointed by the English government.  If the governors had complete power, they would be able to do whatever they wanted to the colonists.

However, the colonists did have the "power of the purse."  The governors' salaries and many other things were paid for by taxes and such that were imposed by assemblies in each colony.  The assemblies were made up of people elected by the colonists.

This is, in effect, a check on the power of the governors.  If the colonists really didn't like how the governors were acting, they could just stop paying them.  In this way, the colonists kept some control over the governors.