The first thing Brutus mentions about Caesar's ghost when he is on the battlefield at Phillipi is the following:

O Julius Caesar, thou art mighty yet!
Thy spirit walks abroad and turns our swords
In our own proper entrails.

Brutus expresses the superstition that Caesar's spirit has come for vengeance and forces his men to turn their swords against themselves, i.e to commit suicide. Later, he once again refers to Caesar's ghost when he says to Volumnius:

The ghost of Caesar hath appear'd to me
Two several times by night; at Sardis once,
And, this last night, here in Philippi fields:
I know my hour is come.

Brutus realises that his time has come. He believes that the appearance of Caesar's ghost the previous night, prior to their battle at Philippi, and again on the battlefield, was an ominous sign that he would die. Whilst he had been in his tent at Sardis, Caesar's spirit appeared:

... Ha! who comes here?
I think it is the weakness of mine eyes
That shapes this monstrous apparition.
It comes upon me. Art thou any thing?
Art thou some god, some angel, or some devil,
That makest my blood cold and my hair to stare?
Speak to me what thou art.
GHOST
Thy evil spirit, Brutus.
BRUTUS
Why comest thou?
GHOST
To tell thee thou shalt see me at Philippi.
BRUTUS
Well; then I shall see thee again?
GHOST
Ay, at Philippi.
BRUTUS
Why, I will see thee at Philippi, then.

Brutus was clearly shaken by the appearance of Caesar's spirit and enquired of his other men why they had cried out. He believed that he may have imagined what he had seen and that his men had been crying out in their sleep. However, all of them denied having made a sound, making it more convincing that what he had seen was real.

It has become obvious to Brutus that the ghost had kept its promise. His troops have been overrun and there is no way out. The victorious Antony and Octavius are coming ever closer. His men ask him to flee but Brutus decides to commit suicide, just like Cassius has done. He runs onto his sword (held by Strato) and kills himself, stating in his last breath:

Caesar, now be still:
I kill'd not thee with half so good a will.

He means that he had a greater desire to take his own life than he had in taking Caesar's. Caesar's ghost can now come to rest because its vengeance is complete.

What does Brutus say about Caesar’s spirit at Philippi?

In Act 4 Scene 3, Caesar’s ghost appears to Brutus, telling him first that he is “thy evil spirit” (325) and second that he shall see Brutus “at Philippi” (138).  Brutus responds, “Why, I will see thee at Philippi, then” (330), in a tone that is open to interpretation, but I would argue he is afraid, for only after the ghost disappears does Brutus says he would have liked to talk to him more, but “now I have taken heart, thou vanishest” (331), as if the ghost were afraid of him instead of the other way around. He’s fooling himself, indicated by the fact he immediately calls Lucius, hoping he had made the noise or had seen the apparition, but of course Lucius had not. He accuses Varro and Claudius of the same thing:  Have they called out in their sleep, he asks? Have they seen anything? They protest they have not, and with that the act ends.  People in Shakespeare's time understood ghosts could be good or evil, and the fact that this one proclaimed itself the latter does not bode well for Brutus. After Cassius dies at Phillip and he understands he is doomed, Brutus again responds to Caesar, telling him he is "mighty yet," for his "spirit walks abroad and turns our swords / In our own proper entrails" (5.3.105-108), referring to the fact that after learning Cassius is dead, Titinius kills himself on his comrade's sword. Brutus calls his fallen comrades "the last of all the Romans," indicating his deep feelings for them.