Renly was more likable than Stannis as was shown in the series. Renly had the backing of Highgarden and most of Storm's End as well. He had the largest army and there was taking the crown by force, much like Robert had.
Renly: No one want you for their King. You never wanted any friends, brother, but a man without friends is a man without power.
Stannis: You think a few bolts of cloth will make you King?
Renly: No, the men holding those bolts of cloth will make me King.
And to reinforce what is seen on the show (I thought a similar line was said, but haven't found a clip...), while Renly is speaking to Catelyn about Robb and the North's rebellion:
Renly shrugged. "Tell me, what right did my brother Robert ever have to the Iron Throne?" He did not wait for an answer. "Oh, there was talk of the blood ties between Baratheon and Targaryen, of weddings a hundred years past, of second sons and elder daughters. No one but the maesters care about any of it. Robert won the throne with his warhammer." He swept a hand across the campfires that burned from horizon to horizon. "Well, there is my claim, as good as Robert's ever was. If your son supports me as his father supported Robert, he'll not find me ungenerous. I will gladly confirm him in all his lands, titles, and honors. He can rule in Winterfell as he pleases. He can even go on calling himself King in the North if he likes, so long as he bends the knee and does me homage as his overlord. King is only a word, but fealty, loyalty, service . . . those I must have." - A Clash of Kings - Catelyn II