Game of Thrones question- How does Robert Barathian come to power?

I watched the first season of "Game of Thrones" on HBO and listened to the book on tape, but sometimes I miss little important details, and with Game of Thrones, there are a lot of those (and so many characters and plot lines to keep straight.) My question is this: How did Robert Barathian become King if the Targarian's sat on the throne for many generations? What was his claim to the throne? And if Jaime Lanaster killed Aegon Targarian,(the last king,) and comes from a powerful family, why didn't he try to take the throne for himself or his father?Game of Thrones question- How does Robert Barathian come to power?
if i remember correctly while Jaime struck the fatal blow, the revolution was led by Robert, Ned, and John Arron so on some level credit goes to the leader



also I think Robert dropped a line saying that the physical act of sitting on the Throne also gave the new king a certain level of credibility... he mentioned that Ned had the choice of assuming the throne but kept it empty waiting for Robert to sit down -- whether Ned physically stopped others I don't know, but at the very least he opted out



also another point is that Jaime was the Hand of the King, had he assumed the Throne there might have been issues with his Motivation and definition as a Traitor (its been implied if not directly stated that he stayed Loyal as long as it suit him, and only killed Targaryen when he realized it was politically savvy to do so. assuming the Throne would have called attention to his motives and to another extent his personal life (as far as I know not even Tywin -his father- knows he's shagging Cersei)



Robert didn't keep him on as Hand for a reason... so no doubt the same reasoning from the populace would have shortened his reign had he tried to become King