When answering this question, you can't leave out the situation at hand--the Misfit is about to kill the Grandmother. That's pretty intense--if you were in that situation, the emotions would be heightened, and you'd probably be begging and crying just like she was. This automatically intensifies things, and brings the two characters very, very close, as they parry back and forth in the dramatic situation. So it isn't that they have some sort of back-story, or personal relationship or anything, they are just in a really intense situation.
Something else that is "going on" between these two characters is a religious discussion. The Grandmother, desperate to live, brings up mercy, and Jesus Christ, hoping to appeal to the Misfit's softer side, to let him know that Christ redeems all men, even bad ones. This was the wrong approach to take, as the Misfit picks up on that thread and discounts it, saying that by raising the dead Christ had "thrown everything off balance," and that he didn't take any stock in Jesus, but in living life to its fullest by being mean. He gets in her face saying he wished he had been there when Christ raised the dead, because then "I wouldn't be like I am now." She sees his weak spot, and in a revelatory change of character, the Grandmother proclaims with love that he was "one of my own children," either speaking from God's point of view, or from some unconditional love that is washing over her at the moment.
So to intensify an already taught situation, they argue over God and Christ, and the Grandmother seems to have some sort of pre-death revelation about love. Emotions are running high, they are debating the soul's worth and value, and lives are at stake. All of these things make for a quickly forged, almost intimate relationship between two rather random characters. I hope that those thoughts helped; good luck!