In Macbeth, what does the phrase "heat oppressed brain" mean?
In this passage, Macbeth sees a dagger before him and wonders if it is merely a figment of his imagination, a projection of what is going on in his own mind. If that is so, the dagger is nothing more than the product of his "heat-oppressed brain." He calls his brain "heat-oppressed" in recognition of his own intensely heightened emotional state. He understands that, being in a state of extreme desire—in this case, to become king—he may very well not be thinking straight and may therefore be deluding himself.
In this passage, Macbeth, like Hamlet when he sees the ghost, doesn't know if a vision he sees before him is real or simply his imagination. Is there another world of real signs and portents, as indicated by the witches? Is the dagger he may or may not see a sign from the other side, the supernatural world, indicating that he should proceed? We know it is not his own dagger because at the end of the passage he says that the dagger before him is as "palpable" (real) as the one "which now I draw," meaning that he is now taking out of its sheath his own, real dagger:
Is this a dagger which I see before me,The handle toward my hand? Come, let me clutch thee.I have thee not, and yet I see thee still.Art thou not, fatal vision, sensibleTo feeling as to sight? Or art thou butA dagger of the mind, a false creation,Proceeding from the heat-oppressèd brain?I see thee yet, in form as palpableAs this which now I draw.
The phrase comes from the end of Act II scene 1, where Macbeth ponders killing Duncan. He clutches a dagger, and proceeds to consider and examine his thoughts about murder, the dagger itself becoming the physical embodiment of the act. He then questions himself, "Is this just a dagger, or is this murder?" Or in other words, "am I really going through with this act?" and these heavy thoughts cause and continue to cause more heavy thoughts, originating from a "heat oppressed" (headache?) brain.
Later in the soliloquy, he mentions the phrase "heat of deeds," heat in the passage suggesting passion; he's therefore not thinking logically, but reacting emotionally to the murder he's about to commit.
In the sixteenth century, people believed that illness, including and perhaps specifically psychological illnesses, were caused by an imbalance of humours in the body. The humours related to combinations of heat-cold and dry-wet.
It may be that Macbeth is suggesting that he has a "choleric" disposition (ie hot and dry) which would be characteristic of leaders kings and generals. To be "oppress'd" by a choleric humour, that is to have an imbalance of humours, would suggest that he has become overly angry, vengeful. The fantasy of the dagger that macbeth thinks he sees, the hallucinations would all be consistent with such an imbalance.