This isn't based on any official information, just pure armchair speculation... but:
On one level, the liquid mirror is just a manifestation of the dissolution of the fake reality of the Matrix. It evokes a dreamlike quality in keeping with Morpheus' lines during the scene, and also provides an element of growing tension that leads up to the shock of waking up.
This, however, doesn't explain why it's specifically a mirror — a melting wall would do just as well for that. On another level, though, a mirror is a metaphor for both perception and for a boundary. (Note the focus on Morpheus' mirror shades, and on Neo's reflection in them, in the previous scene.)
Before taking the pill, Neo's perception of himself and his surroundings was flawed, like the broken mirror — he perceived the illusion of the Matrix as real. The healing of the mirror is the first step in his beginning to perceive the Matrix for what it really is.
By touching the surface of the mirror, Neo takes a step across the boundary between the Matrix and the real world (there's almost certainly an allusion to Lewis Carroll's Through the Looking-Glass here, just as the previous scene referenced Alice in Wonderland), a step whose irreversibility is shown by the mirror swallowing up Neo completely.
On yet another (less metaphorical) level, it's not really clear whether the mirror engulfing Neo actually represents an effect of the pill, or of the Matrix fighting against it. In particular, some of the other characters' exclamations while the process is occurring could be taken to imply that the mirror represents some kind of countermeasure that is trying to kill Neo before the others can free him.
Then again, that ambiguity may well also be intentional — there's a strong element of death and rebirth in the scene, and in the one that follows it. It might not matter whether it's the Matrix or the pill that's trying to kill Neo; the important thing is that he dies in the Matrix and is reborn in the real world.