When Neo is shown the world as it really is, he seems to struggle with the revelation, jacks out and then collapses to the ground. Cypher says ...
He's gonna pop!
... and Neo is sick all over the floor.
What is "popping"?
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An earlier version of the script (dated April, 1996 and available online here) indicates that "popping" is a possible negative consequence that can occur when a person, especially an adult, is released from the Matrix. They literally die as a result of the shock caused by seeing/learning the truth of their existence.
CYPHER: Morpheus, I know what you believe but I think this is a mistake. We're rushing him.
He's old. I'm afraid he might pop.
MORPHEUS: Haven't I always told you, Cypher, not to let fear control your life.
Apoc, are we on-line?
This process appears to be akin to a full-body seizure with a simultaneous rise in intercranial pressure and a violent heart attack.
[On his hands and knees, he reels as the world spins. Sweat pours off him as a pressure builds inside his skull as if his brain had been put into a centrifuge.]
CYPHER: He's going to pop!
TRINITY: He's going into arrest!
CYPHER: He's gonna pop!
We actually see this occurring in the Matrix webcomic Saviors. The lead character, Hitch extracts his parents but they die almost immediately from shock.
It would appear to just mean that he was going to throw up, which is precisely what happens. However, the script also indicates that he had a massive amount of pressure building up inside his head which could also be an explanation for what is happening here. The pressure builds up as he comprehends what is happening, then is suddenly released (popped) in the form of throwing up.
INT. MAIN DECK
His eyes snap open and he thrashes against the chair, trying to rip the cable from the back of his neck.
NEO: Get this thing out of me!
TRINITY: Easy, Neo. Easy.
Dozer holds him while Trinity unlocks it. Once it's out, he tears away from them, falling as he trips free of the harness.
NEO: Don't touch me! Get away from me!
On his hands and knees, he reels as the world spins. Sweat pours off him as a pressure builds inside his skull as if his brain had been put into a centrifuge.
NEO: I don't believe it! I don't believe it!
CYPHER: He's going to pop!
Vomiting violently, Neo pitches forward and blacks out.
On a side not "He's gonna blow/pop!" has always been a common saying for "He's about to throw up!" where I'm from, so I immediately assumed that was the case here.