It has to do with the climate... our middle-class consciences ... the Christmas story ("no room in the inn").
The passage cited to support the homeless-Jesus image is most often Matthew 8:20: "Foxes have dens and birds have nests, but the Son of Man has no place to lay his head."
Jesus said this in response to a man indicating that he was willing to follow Jesus wherever he went. It is actually part of a couplet of encounters between men with excuses - either voiced or implied - that Jesus challenges who don't -seemingly - end up following him.
The point in favour of the homeless-Jesus picture painted by this one encounter is that it does occur in the Galilean region. In fact it occurs in Matthew immediately after Jesus was specifically in Capernaum healing Peter's mother in law.
So if we are to take Jesus' words at face value then one is left asking the question "How can Jesus claim to have no where to lay his head, if he occupies/owns a home just around the lake from where he makes the statement?"
The homeless-Jesus idea is powerfully portrayed in Canadian sculptor Timothy Schmalz's statute which sits outside Regis College on the sidewalk in Toronto, Ontario.
Lots of pastors - of course - have taken this image to heart and go spend a night on the street around Christmas and then use their experiences to talk to their congregations about the realities of poverty in their community. Matthew 25 powerfully resonates with this idea of Jesus being the one we truly serve when we visit the imprisoned, feed the hungry, clothe the naked or care for the sick.