Please explain the following quote from Julius Caesar.
"And since you know you cannot see yourself
So well as by reflection, I, your glass
Will modestly discover to yourself
That of yourself which you yet know not of."
I have re-written the quote (from Act I, scene ii) that you ask about. It is important to learn the proper format for reprinting verse quotes from Shakespeare's plays. If you would like to refrain from setting the quote in its proper line structure, then it must be written as follows:
And since you know you cannot see yourself/So well as by reflection, I, your glass/Will modestly discover to yourself/That of yourself which you yet know not of.
This format is key to retaining the structure of the lines of verse, including the capitalization of the words that begin each line.
Now to the meaning of the line. It is spoken by Cassius to Brutus as he begins his seduction of Brutus towards joining the conspiracy against Caesar. Brutus has already mentioned to Cassius in line 38 that he is "[v]exed. . .[o]f late with passions of some difference," suggesting that he too, worries about Caesar being offered a crown.
In the line you have quoted, Cassius plays upon Brutus' internal strife, suggesting that he "knows" Brutus so well that he, by talking out loud about the things that Brutus is unwilling to mention, will show Brutus his own self. Cassius is telling Brutus that, acting as his mirror, he can describe Brutus' inmost thoughts about Caesar and the state of affairs in Rome before Brutus is even aware that he thinks them.
For more on this "seduction" scene, please follow the links below.
Shakespeareguru is absolutely correct. This is one of Cassius' most manipulative and clever techniques to get into Brutus' psychology. Cassius does not let Brutus speak about who he is, rather, he will tell Brutus what to think about himself.