The subject of Bran's assassination attempt is something Game of Thrones (2011) has not answered, so one must turn to the books to really understand. Even the books do not outright admit who commissioned the assassination, but they sternly imply it was Joffrey who sent the killer to Winterfell armed with Tyrion’s dagger.

Tyrion concluded after his own investigations that it was his nephew. Joffrey overheard King Robert admitting to Cersei that putting Bran out of his misery would be the merciful and brave thing to do. Joffrey, being a poor assassination planner, stole a recognizable and traceable dragonbone dagger, hired an assassin, and armed him with orders to dispatch Bran.

Also, a conversation between Cersei and Jamie from A Storm of Swords rather clearly identifies Joffrey as the culprit in the eyes of his true parents. Cersei begins the exchange:

“Yes, I hoped the boy would die. So did you. Even Robert thought that would have been for the best. ‘We kill our horses when they break a leg, and our dogs when they go blind, but we are too weak to give the same mercy to crippled children,’ he told me. He was blind himself at the time, from drink.”

Robert? Jaime had guarded the king long enough to know that Robert Baratheon said things in his cups that he would have denied angrily the next day. “Were you alone when Robert said this?”

“You don’t think he said it to Ned Stark, I hope? Of course we were alone. Us and the children." Cersei removed her hairnet and draped it over a bedpost, then shook out her golden curls. "Perhaps Myrcella sent this man with the dagger, do you think so?"

It was meant as mockery, but she'd cut right to the heart of it, Jaime saw at once. "Not Myrcella. Joffrey."

Cersei frowned. "Joffrey had no love for Robb Stark, but the younger boy was nothing to him. He was only a child himself."

"A child hungry for a pat on the head from that sot you let him believe was his father." He had an uncomfortable thought. "Tyrion almost died because of this bloody dagger. If he knew the whole thing was Joffrey's work, that might be why..."

The truth thus never comes out, because Cersei and Jamie aren't going to tell anyone, Joffrey, Catelyn and the assassin are dead, and Bran never witnessed anything.

Joffrey's motivations are unclear and are widely theorized. Some believe it was an expression of his father issues and he did it to impress Robert. Some believe it was simply because he enjoyed causing harm by proxy, which he had been shown doing many times. Finally, it could have been revenge for the embarrassment Bran caused Joffrey on his first trip to Winterfell early in the story.