The title of the play means that each person is only six social steps away from another person. In other words, if you think of anyone in the world, there are only six people between you and that person through social connections like work, school, and family.
In John Guare's play, Paul uses social connections and behaviors to play the part of someone who is wealthy and connected. He uses social tactics like appearing a certain way, mentioning a family member, and wearing specific clothes to convince a couple that he knows their children and belongs in their social sphere. He is able to do this by learning their information and behavior from their children.
One of the interesting feelings the play evokes is the idea that we're all simultaneously connected to each other and disconnected from each other. Though people may be only six social connections apart, they are separated by large gulfs of behaviors, lifestyles, and groups that human society creates. Paul went to school with the children of the older adults in the play but is still not considered one of them—not really.
In the end, the characters' lives are transformed—for better or worse—by their connections with Paul. Even though they don't know his true name or what happens to him, he has made an impact on their lives. As for Paul himself, it seems that what he wanted most of all was to have tighter connections with these people and for them to see him as family.