What does it mean when Benvolio said to Romeo "I will show you shining at this feast" in Romeo and Juliet?
Benvolio is telling Romeo that there are better girls for him if he will just look around for them.
Romeo is upset because Rosaline does not return his feelings. He is in a deep depression, feeling that there will never be another girl for him. His cousin Benvolio tries to convince him otherwise. Mercutio and Benvolio are planning to go to the Capulet ball, and they think Romeo might meet a girl there who will turn his head.
But in that crystal scales let there be weigh'd
Your lady's love against some other maid
That I will show you shining at this feast,
And she shall scant show well that now seems best. (Act 1, Scene 3)
Romeo is skeptical. He agrees to go to the party. When he saw the invitation list and realized that Rosaline might be there, he hoped to catch a glimpse of her. Instead, Romeo sees Juliet and she completely changes his mind. He earlier spoke of Rosaline's incomparable beauty, but when he sees Juliet he is stunned.
O, she doth teach the torches to burn bright!
It seems she hangs upon the cheek of night
Like a rich jewel in an Ethiope's ear;
Beauty too rich for use, for earth too dear! (Act 1, Scene 5)
Romeo’s head was turned. Benvolio was right. He forgets completely about Rosaline. He basically pines for Juliet from then on, even climbing into the Capulet orchard to see her. He does not care that she is a Capulet and he is a Montague. He is determined to win her heart.
Juliet seems to fall for Romeo too. She decides to risk it, because she personally has nothing against him and his name does not make him who he is. He may be her mortal enemy, but she fell in love with him at first sight and the rest is history.