How does Bassanio know that he is a welcome suitor?

In Shakespeare's play The Merchant of Venice, Bassanio is in love with the beautiful lady Portia. He feels that he must make expensive gestures to win Portia's love, not realizing that she already loves him. Portia is speaking to her maid, Nerissa, early in the play about how she dislikes all of her current suitors and how much she enjoyed a previous visit from Bassanio:

NERISSA: Do you not remember, lady, in your father's time, aVenetian, a scholar and a soldier, that came hitherin company of the Marquis of Montferrat?PORTIA: Yes, yes, it was Bassanio; as I think, he was so called.NERISSA: True, madam: he, of all the men that ever my foolisheyes looked upon, was the best deserving a fair lady.PORTIA: I remember him well, and I remember him worthy ofthy praise.

Meanwhile, her current suitors try to win Portia's hand by passing a strange test her father mandated in his will: anyone who wishes to marry Portia must choose one of three caskets (gold, silver, and lead). If he chooses the casket which contains Portia's portrait, he can marry her; otherwise, he must leave. No suitor has yet passed this test, and there are some scenes showing suitors picking the wrong caskets (gold and silver). Portia is very tired of this whole charade, so when Bassanio arrives at her house to declare his love and try the test, she is sad, as she doesn't want to lose the possibility of marrying him:

PORTIA: I pray you, tarry: pause a day or twoBefore you hazard; for, in choosing wrong,I lose your company: therefore forbear awhile . . . I would detain you here some month or twoBefore you venture for me.

Bassanio nevertheless is determined to try, saying "Let me choose / For as I am, I live upon the rack," tormented by his love for Portia. Portia very reluctantly agrees to let him try, but "with much, much more dismay / I view the fight than thou that makest the fray."
To their joy, Bassanio passes the test by choosing the casket with Portia's portrait inside—the lead casket which the other suitors automatically passed over in favor of the gold and silver caskets. Bassanio is not taken in by the more expensive caskets, indeed he reasons:

ornament is but the guiled shoreTo a most dangerous sea . . . The seeming truth which cunning times put onTo entrap the wisest. Therefore, thou gaudy gold,Hard food for Midas, I will none of thee;Nor none of thee, thou pale and common drudge [silver]'Tween man and man: but thou, thou meagre lead,Which rather threatenest than dost promise aught,Thy paleness moves me more than eloquence;And here choose I; joy be the consequence!

Bassanio is not deceived by appearances but believes true beauty is more than mere outward spectacle. He is correct, and he wins Portia's hand in marriage.
http://shakespeare.mit.edu/merchant/full.html