First, let's have a general definition of liberal humanism. Broken into the two terms, humanism is obviously a focus on human affairs and liberalism is a focus on individual freedom and agency. Liberal humanism is a philosophy that emphasizes the value and function of individual thought and empirical knowledge; rather than relying on a religious or metaphysical doctrine. Liberal humanism is not necessarily secular (there is a philosophy of religious humanism), but the idea is that liberal humanists are interested in creating and supporting conditions which allow the individual to live and make choice according to his/her will.
Through a Christian lens, it is possible that the grandmother has a moment of divining God's grace; that she realizes she and the Misfit are both sinners, that this moment is only made possible by a divine understanding. And, in a religious frame, this makes sense. The grandmother had proven herself to be a racist, selfish person throughout the entire story. The Misfit has fallen from grace and is comfortable with that fact. There is nothing to indicate, at this moment of grace, that the Misfit (perhaps on the verge of crying) should have a moment of remorse. There is also nothing to indicate that the grandmother is really connecting with the Misfit, being selfless; if anything, it would seem that she says this to the Misfit to connect with him only to save her own life. That being said, a metaphysical or spiritual epiphany would have to occur for the grandmother (and/or the Misfit) to experience a real change of heart.
But, a liberal humanist would dismiss that epiphany/moment of grace. The Misfit calmly admits that he is no good. He indicates that if he would have received some type of grace or direction by God, he would not be the bad person that he is. Concerning Jesus raising the dead, he says:
"Listen, lady," he said in a high voice, "if I had of been there I would have known and I wouldn't be like I am now."
Here, the Misfit can not rely on faith. He expresses an empirical perspective, saying that if he'd had proof of God when he was younger, he would not be the way he is. Therefore, he would have required a real, physical demonstration, not some doctrine of faith nor some metaphysical notion about goodness. One could imply that the Misfit simply needed direction; human direction and compassion when he was younger. Here, the Misfit expresses a liberal humanist perspective: he needed to see the actual event of Jesus raising the dead. Faith in the act is not enough.
The Misfit says that grandmother also needed an actual, physical event to make her a good person: "She would have been a good woman," The Misfit said, "if it had been somebody there to shoot her every minute of her life." The grandmother claimed to be religious and from "good" people, but her faith was not enough. She needed an actual physical consequence to even suggest that she might make a selfless, meaningful connection to another person. The liberal humanist would agree, saying that these people acted and became who they are based on real social conditions and individual actions.