Originally Posted by Son of Coul
They connected on an intellectual level, plus Stark got a kick out of a meek little nerdy dude unleashing something so big and destructive as Hulk. He seemed fascinated by the whole concept, believing that deep down a part of Hulk and his rage came from Banner.
And Banner appreciated someone treating him like a normal scientist, as well as addressing his Hulk alter ego with straightforward wry humor, as opposed to tiptoeing around the subject in fear as most tend to do.
There's more to it of course, but I'd say that's the gist of their bromance.
A lot of their connection stemmed from the fact that Tony identified with Bruce's dilemma on a deep level. Tony & Bruce shared a kinship that went beyond their roles as scientists, which Whedon skillfully examined. As Tony pointed out, both of them had near-death experiences that left them irrevocably changed both physically and emotionally. Tony was kidnapped and nearly killed, left with shrapnel near his heart, an arc reactor in his chest and a suit of armor that made him one of the most powerful beings on the planet. Similarly, Bruce nearly died in his gamma radiation accident/experiment and was left with a raging beast inside that made him both extremely powerful and extremely dangerous. Their experiences were two sides of the same coin, which Tony helped Bruce to understand and accept.
One purpose of their relationship was to allow Tony to open Bruce's eyes to the possibility that "The Other Guy" could be a force for good if only he would let go of his fear and embrace all aspects of his being. When he told Bruce to "strut," he wasn't just trying to get him to "hulk out," but encouraging him to take ownership of the Hulk and direct his power towards good ends. Tony's belief in Bruce helped Bruce believe in himself. That is why Tony knew Bruce would come back for the Battle of New York even when everyone else, even Bruce himself, doubted that he would.
The writing and acting were very subtle and never beat the audience over the head with the reasons for their relationship, for which Whedon, Downey and Ruffalo should be congratulated.