Jonathan Edwards’s purpose in delivering the sermon, "Sinners in the Hands of an Angry God" is to warn his congregation in particular, and presumably, by extension, his nation as a whole, that they must repent of their sinful ways and turn to God for forgiveness before it is too late - so that they can escape death by hell fire.
In essence, Jonathan Edward's is saying that the opportunity is now to embrace and accept Jesus Christ, his message, his teachings, and his sacrifice as atonement for sins. The choice must be made by those God is calling, through Jesus Christ, to either accept the call to repentance and a new life, or reject it.
Rejection will result in death by hell fire. Edwards indicates that God is extending mercy by calling out to sinners - He is giving them an opportunity to accept Jesus' sacrifice and start a new life of obedience to Christ - to become slaves of Christ - subject to his mastery over them. Edwards' purpose with this sermon is to awaken people to the seriousness of their situation because of sin - and that Christ is their only way to true redemption and a real eternal life. Edwards is encouraging his congregation to flee,run, from the wrath that is to come upon the unrepentant and unbelieving. He trumpets the necessity of accepting Christ and his atoning death. This is indicated in the Scriptures as well in this verse:
"that at the name of Jesus every knee should bow, in heaven and on earth and under the earth,"
(Philippians 2:10 - New International Version (NIV) )