Very simple, they assumed he was guilty and convicted based on circumstantial evidence. No actual case was built from the on sight analysis of the situation. The point of Coffey being on death row was a commentary on society being undeserving of a true gift from God, in fact there was a not so subtle reference prior to the execution that Tom Hanks' character did not want to face God after destroying one of his blessings. He wasn't a coward for choosing death, he was hurting from the pain that humanity was causing him, and the "golden rule" came into play, he didn't hurt anyone who was slightly flawed, but rather one sick human being that tortured innocents and the criminal that killed the two children.
You don't know what circumstantial evidence means and there was no commentary on the legal system at all.
My main function as a Christian is to be the biggest possible jerk as a demonstration of the awe found in God's Grace and Forgiveness. (That love part comes in to play at some point)