Why was the outcome of the French and Indian War significant to the colonists?

The defeat of the French in North America removed what had previously been a long-standing threat in American life. While the French possessions were very sparsely settled compared to British North America, they posed a continual threat in times of war, especially because of the indigenous allies the French had. The defeat of the French meant that, for one of the very first times in the American colonies, the possibility of attacks by the French was effectively eliminated. This had the effect of removing one of the ties to Britain, because the American colonies were suddenly less dependent on Britain for defense, even as British troops remained stationed in North America.
More seriously, the war left Great Britain massively in debt. Part of its war effort in North America had depended on generous subsidies to colonial legislatures, particularly the New England colonies, to raise militias and keep them in the field year after year. That had enabled the eventual defeat of the French, but by 1763, Great Britain's national debt had gone from 75 million pounds to 133 million pounds. Part of that had been a consequence of the larger war in Europe (remember, the French & Indian Wars were just one theater of the Seven Years War in Europe), but many British politicians felt that the American colonists needed to help pay back the sizable debts that had been incurred.
This combined with the need to pay for the troops who were still stationed in North America led to the passage of laws, such as the Stamp Act, that were a direct tax to raise revenue for Britain. This entirely circumvented colonial legislatures, who now complained that they were being taxed without representation. While the Stamp Act was repealed after substantial protest, it stirred colonial American grievances over British tax policy and fostered a sense of unity among the aggrieved colonists.