"Two chairs appear in front of them, and Neo reaches out to touch one. He runs his hand along it's back."
The sudden appearance of the chairs indicates their illusionary nature. Neo is deceived by the appearance because he "reaches out to touch one", even though it is merely the output of a computer program that is projected into Neo's consciousness.
Of course, Neo's image himself is an appearance as well, as indicated previously. The only way he can make sure that he is a real existence rather than a program is by checking his thinking process and self-awareness. This is René Descartes's "Cogito ergo sum".
The Matrix is a thought experiment to convey the problem that everything that we perceive might be an illusion. Some other similar thought experiments are: Hillary Putnam's thought experiment of Brains in a vat and Plato's Allegory of the Cave.
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