In which play by Shakespeare can I find this quote, "When I saw you I fell in love, and you smiled because you knew"?.

This quote is from the Opera Falstaff by Verdi.  Fenton sings it to Nannetta in Act 2 Part 2: 

Come ti vidi
M’innamorai,
E tu sorridi
Perchè lo sai.

The Italian can be translated to:

"When I saw you I fell in love, and you smiled because you knew."  

It is a lovely opera.  Bioto (mentioned by previous users) who wrote the libretto for the opera plagiarized a lot of quotes from Shakespeare: Henry IV, Romeo and Juliet, Merry Wives of Windsor, and Othello, among others.  This particular line does not exist in ANY of Shakespeare's works.  ( I personally think it is funny that Bioto plagiarized Shakespeare and one of the few lines he actually made up-- credit is given to the Bard.)

So why the confusion?

Blame Facebook and Twitter-  There is also some blame to go to a small print shop that printed it on throw pillows back in the early 2000s and attributed it to Shakespeare.  Truth be known, someone probably found the quote, saw it was from Falstaff, and assumed that Shakespeare wrote it, never realizing that Shakespeare never named a play after one of his most famous characters.