He's saying that all the parts of a snake cannot live without one another. The tail (S.C = South Carolina) needs the head (N.E. = New England), the head needs the next part down and so on. So in other words, Franklin is saying that the states cannot survive on their own and they must all join together forming the United States.
His audience was colonists who were unsure of whether to join the rebellion against the British. At the time, many people were loyal to England as they just came over the Atlantic from there. Though over time, various factors pushed the two sides away to lead to the Revolutionary War.