What is the reason for the letter that Cassius wrote in Shakespeare's Julius Caesar?

Cassius wrote a number of letters to Brutus. He used different handwriting in each letter to make it look like different Roman citizens wrote the letters. The letter given to Brutus by his servant Lucius in Act II, Scene 1, which he partially reads aloud, is only one of the many Cassius concocts and plants in various places to try to convince Brutus to become the leader of a conspiracy to assassinate Julius Caesar. It can be assumed that all of the letters Cassius writes for Brutus have the same general theme. Brutus himself enunciates that theme after he finishes reading the letter given to him by Lucius.

"Shall Rome, et cetera?" Thus must I piece it out:
Shall Rome stand under one man's awe? What, Rome?
My ancestors did from the streets of Rome
The Tarquin drive when he was called a king.
"Speak, strike, redress." Am I entreated
To speak and strike? O Rome, I make thee promise,
If the redress will follow, thou receivest
Thy full petition at the hand of Brutus.

It is because of Brutus' distinguished ancestry that Cassius and other Romans are so anxious to draw him into their conspiracy. They are thinking about assassinating Caesar and the problems that will arise if the assassination is successful. Brutus can help them justify the assassination to the populace and will serve as an ideal head of the new government that will have to be established. Cassius understands this better than anyone.

Shakespeare could not show Cassius writing a lot of letters or Brutus reading a lot of letters because it would take up too much time. At the end of Scene 2 in Act I, Cassius tells himself, and the audience, what he intends to do to follow up on the strong pressure he has been putting on Brutus to join in his conspiracy against Caesar.

I will this night,
In several hands, in at his windows throw,
As if they came from several citizens,
Writings, all tending to the great opinion
That Rome holds of his name, wherein obscurely
Caesar's ambition shall be glanced at.
And after this let Caesar seat him sure;
For we will shake him, or worse days endure.

The letter Brutus reads aloud is one Cassius wrote and threw in his windows. There was no mail-delivery service in ancient Rome; those few prosperous citizens who knew how to read and write must have usually had their letters hand-delivered by servants. Cassius may be delivering his own letters to make sure there are no slip-ups in his scheme to deceive Brutus.

In Act I, Scene 3, Cassius gives some additional letters to Cinna and tells him:

Good Cinna, take this paper,
And look you lay it in the praetor's chair,
Where Brutus may but find it; and throw this
In at his window. Set this up with wax
Upon old Brutus' statue. All this done,
Repair to Pompey's Porch where you shall find us.