The short answer is that we still don't know who sent the catspaw from show canon and so we don't know how the catspaw ended up using the Valyrian steel dagger. However, what we do know that is whoever sent the catspaw gave him the dagger, probably as payment, because of how expensive such an item is. Until it is revealed in show canon who sent the catspaw this is simply left as we don't know.
If we were to look at book canon, however, we find out that it was almost certainly Joffrey that sent the assassin and that he gave him the blade because he was:
- Unaware of what Valyrian steel is
- The blade is actually quite plain looking (or at least per book canon, in the show it isn't)
The answer to your question from book canon is essentially Joffrey wanted to please his father and so sent the assassin after Bran as a "mercy killing". To do this he chose a dagger from Robert's collection and gave it to the assassin to do the deed with.
He remembered a cold morning when he'd climbed down the steep exterior steps from Winterfell's library to find Prince Joffrey jesting with the Hound about killing wolves. Send a dog to kill a wolf, he said. Even Joffrey was not so foolish as to command Sandor Clegane to slay a son of Eddard Stark, however; the Hound would have gone to Cersei. Instead the boy found his catspaw among the unsavory lot of freeriders, merchants, and camp followers who'd attached themselves to the king's party as they made their way north. Some poxy lackwit willing to risk his life for a prince's favor and a little coin. Tyrion wondered whose idea it had been to wait until Robert left Winterfell before opening Bran's throat. Joff's, most like. No doubt he thought it was the height of cunning.
The blade Joff chose was nice and plain. No goldwork, no jewels in the hilt, no silver inlay on the blade. King Robert never wore it, had likely forgotten he owned it. Yet the Valyrian steel was deadly sharp . . . sharp enough to slice through skin, flesh, and muscle in one quick stroke. I am no stranger to Valyrian steel. But he had been, hadn't he? Else he would never have been so foolish as to pick Littlefinger's knife.
The why of it still eluded him. Simple cruelty, perhaps? His nephew had that in abundance. It was all Tyrion could do not to retch up all the wine he'd drunk, piss in his breeches, or both. He squirmed uncomfortably. He ought to have held his tongue at breakfast. The boy knows I know now. My big mouth will be the death of me, I swear it.
A Storm of Swords, Tyrion VIII
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..."
A Storm of Swords, Jaime IX