What does Jonathan Edwards hope to accomplish with his message in Sinners in the Hands of an Angry God?

Jonathan Edwards' goal was to awaken people to what, to him, was the very real danger of continuing in their sin, to get them to repent and call on God for mercy. Though he intended to scare and sober his audience, his goal was not to drive them away, but ultimately to bring them to God.
Edwards preached the sermon in response to what he sensed was a complacent attitude in the town where he was preaching. People were professing Christians, but they assumed that by attending church, they had done all they needed to do. They were not thinking in terms of God as being a real entity whom they had wronged, and with whom they desperately needed to repair the relationship. Edwards sought to shake them out of their complacency by putting vividly before them just how ugly sin is in the sight of God, and how seriously God takes sin. Only then would they take their own sin equally seriously, enough to repent of it, actually stop doing it, and change their lifestyle while calling on God for help.
Thus, Edwards' use of the words "angry God" was not intended to tell his audience that God had rejected them forever. (Then they would give up, or lash out at God in reciprocal anger.) Rather, he was in essence saying, "Your attitude and lifestyle is angering God ... do something about it!" He points out that when people turn to God in repentance, God will quickly forgive.
Edwards' ideal response to such a sermon from his audience would be for them to repent, with genuine sorrow, over their sin; hate it, desire to leave it, and to beg with God to forgive them and draw them closer to Him, and to do all this, not in a "go-home-and-think-about-it" kind of way, but with a sense of urgency. This is exactly the response that many in the congregation had, even weeping and crying out in the middle of the service.