The question why do "I" think people were persuaded to change their lives as a result of Edwards sermon "Sinners in the hands of an Angry God", is an opinion and I would think there are many opinions with a wide variety of responses. However to uncover some truths one must consider some of the facts which are more religious / spiritual in nature.
Jonathan Edwards was a minister doing the work of God and his manual was the Bible. The Bible story is one of birth, falling into sin, rebirth, and redemption if one will accept the gift of God's son for that redemption. However mankind's nature, due to the transgression in the Garden of Eden cause men to be in a sinful state. A state of self-righteousness, a pride that exalts them above others and allows them to think about self first. Self being on the throne of the heart and not God. Throughout history man has fallen into a depravity of sorts, drinking, gambling, killing. This self destructive behavior is why mankind needs saved from himself.
Many after falling to a very low place in life understand they need help. They want help and they need hope, peace, and love back in their lives. They understand they are not in a good place in life. Enter J. Edwards, he understood that if mankind did not change his ways and turn the throne of their hearts to God, then the result would be an eternity in hell. A place of horror and pain. In Sinners in the Hands of an angry God, those being preached to understood that they were "sinners" and that God would not hold out his hand of mercy forever. They felt the shame of their actions and as we all know life is not forever on this earth, they would have a day of reckoning. When standing before God the choice would be to hear "well done", or be thrown into the lake of fire for ones depravity and the lack of repentance. People were persuaded to change their lives because they wanted to feel hope again. Edwards offered them hope. For those that were changed the idea of no longer being sinners allowed them the hope of heaven at death. The other option, to spend eternity in hell because God was angry for their lack of repentance, well who wants to go there.
Edwards was a powerful speaker in his day and he spent a lot of time praying about the message he brought to the people. The culture and the people understood the polarities in life. It was Heaven or Hell, Right or Wrong, Black or White. The majority of people held the traditional Christian beliefs and the word of God or The Bible was the book to be accountable to. End result many lives changed for fear of going to hell if they did not.