The Prince of Arragon, as his title communicates, was born a royal and has lived all his life with that unearned privilege. We can imagine he has been flattered and catered to all his life and that this has created his sense of importance and inflated self worth. The silver casket says the owner will get what he deserves. The prince's line of reasoning is that he should pick that casket because he deserves the best.
He rejects the gold casket because he thinks that one is too obviously a trick, and the lead is beneath his dignity.
Ironically, the prince, before choosing the silver casket, gives a long speech about merit or worth. If people succeeded based on their merit or skills alone, not the accident of being born to a privileged position, the world would be a different place, he says. Those who give orders would have to take orders. Many of the "low peasantry" would be honored. Yet the prince never stops to thinks that maybe his own sense of deserving the best and being entitled to whatever he wants comes not from his own merit but is just because he was born a prince.
The prince is sure he deserves Portia, but has a rude awakening when the silver casket shows:
The portrait of a blinking idiot
Presenting me a schedule! I will read it.—
How much unlike art thou to Portia!
How much unlike my hopes and my deservings!
The casket calls him a fool and tells him to go away.