Can someone explain how this works? I did a cake for my son's teacher at the end of the school year. Had all the kids signs their names on regular paper and then had that transferred to edible paper. I cut the names out to put them on different places on the cake but it never "melted" into the icing. What did I do wrong?

It depends on the material for the image. Rice paper is most common because it is stronger, but it only softens, it won't melt. Icing sheets are more fragile and do tend to melt in more.

You didn't do anything wrong, it is just the nature of the beast.