Mark G. answered 09/12/23
PhD in Theology with 17+ years of teaching experience
The religious revivals of that time put a great deal of emphasis on the personal experience of God through the Holy Spirit. When Christianity takes a turn toward the authority of individual experience, traditional power structures are disrupted. Protestant Christianity had already disrupted things by encouraging people to read and interpret the Bible for themselves -- not filtered through official Church doctrine as determined by the clergy. Still, many Protestant denominations retained hierarchical authority, with pastors serving as "experts" to help interpret scripture. A personal experience of the Spirit removed all intermediaries (even, to some degree, scripture) between the person and God -- there was no longer any need to test faith against official teachings.
The result was a more general confidence in individuals to make judgments of all sorts for themselves -- including political ones. This change rubbed against the class-based monarchy of Britain and its dependence on belief in a natural hierarchy. Christians came to believe in self-government.
Additionally, many colonial Christians were influenced by the Puritan Revolution in England during the 17th century. In the rhetoric of the Puritan leaders and chaplains, there is a pretty firm belief in their being called by God to rebel against the king and try to implement a radically different political structure. American preachers -- especially the Presbyterians -- promoted the idea of people being called to a revolutionary "vocation".