Homeless people need public services to survive, and those are much greater in big cities, which are much more common in Blue States. The homeless live in cities where they can use libraries, panhandle, live under bridges and other structures to stay out of the weather, and seek shelter at actual homeless shelters in case of extreme weather. They also carry everything with them and therefore don't go far from where they sleep, usually, and can take advantage of public transportation. There is little for you in rural places if you are homeless, except the kindness of extended family. All of these factors together are an adequate explanation for why there are more homeless in big cities, hence blue states.