There were many experiences and traditions that inspired this in the Americans.
As far as traditions go, the most important tradition was the tradition of opposition to royal power in the British political system. This sort of tradition had started as far back as the Magna Carta in 1215 and had continued up through events such as the Glorious Revolution of 1688. This tradition helped make Americans more likely to want an independent and representative government.
One experience that happened in England, but which pushed some colonists towards a desire for independence was the oppression of Puritans because of their dissenting religious beliefs. This inspired many to move to the colonies and led directly to a desire for representative government (and indirectly to a desire for independence).