WHY DO The ELECTED REPRESENTATIVES get to promote a political message on the public streets?
Because they are the elected officials and therefore in charge of the streets and city maintenance. Pretty simple.
What are the guidelines that permit a political message to be displayed on a public road? What makes it legal?
The question is if there is anything that makes it illegal. If not, then it is legal.
I would assume that there probably are city or state guidelines/rules that streets remain safe and traffic indicators/lines remain clear. Aside from that, anything probably goes.
As for messaging, most politicians try not to raise a ruckus or ignite controversies. At least it used to be that way.
USA on the street is pretty anodyne.
Support the Troops a little more controversial, but passes for good patriotism in the warmongering USA.
Deport Illegals is a lot more political and dicey. Inviting controversy and dissension. A
Confederate Flag is going to arouse protests and opposition. Doesn't mean it wouldn't be legal, thought he attention and reactions it gets might make it unsafe. Religious messages such as Go to Church or Obey God! or WWJD? would run up the establishment clause and be a No No.