MORPHEUS: You have the look of a man who accepts what he sees because he is expecting to wake up. Ironically, this is not far from the truth.
In some dreams, there is a certain point when the rational part of you that was strangely absent suddenly returns and you think, "This is nuts. I'm definitely dreaming." At which point you either wake up or just play along with dream (we always try to start flying but it never works). Well, this is exactly what's happening to Neo. The problem is that what he thinks is the dream and what he thinks is real are inverted. The reality he is about to wake up in is, well, out of this world.
NEO: You ever have that feeling where you're not sure if you're awake or still dreaming?
Neo is uh, a little weirded out by someone hacking his computer and mentioning the Matrix and then predicting a visitor. Because this happened right when he awoke, it almost feels like he never actually woke up. It's our first introduction to the question what's real in the world of The Matrix.
CYPHER: I know this steak doesn't exist. I know that when I put it in my mouth the Matrix is telling my brain that it is juicy and delicious. After nine years, you know what I realized? Ignorance is bliss.
As much as we like to hate Cypher we have to admit, he's quite the philosopher. He probably understands the reality of the Matrix better than anyone else. He knows that the difference between the physical world and the Matrix exists only in our perception of the two. People think that because the Matrix exists in a virtual space, and not a physical one, that it somehow has less worth or is inherently less real that the alternative. But when people are ignorant to this fact, they are perfectly content with the virtual reality. If you don't know a master race of machines is growing you like a vegetable and feeding you your pureed ancestors to use you as an energy source, then you don't care.
SPOON BOY: Do not try and bend the spoon. That's impossible. Instead, only try to realize the truth.
NEO: What truth.
SPOON BOY: There is no spoon.
NEO: There is no spoon.
SPOON BOY: Then you'll see it is not the spoon that bends, it is only yourself.
The spoon boys suggest that reality is completely subjective. If Neo bends then the spoon bends; he is only manipulating himself. In fact, as again the wise spoon lad points out, there is no spoon to manipulate in the first place. The Matrix is simply a series of electrical signals fed into and interpreted by Neo's brain. To change something in the Matrix is merely to change one's own mind. It's so deep we might need to ditch the spoon for a ladle.
NEO: [touching the armchair] This—this isn't real?
MORPHEUS: What is real? How do you define real? If you're talking about what you can feel, what you can smell, what you can taste and see, then real is simply electrical signals interpreted by your brain.
Ah, we've reached the crux: one of the big questions of the movie. Morpheus is explaining to Neo that what makes something real is our perception of it. When we're talking about a physical object that our sensory perception can detect and that we can interact with, then the Matrix isn't any less real than the "real" world. Sure, the armchair Neo is touching has no volume or mass in this other world, but in the world of the construct it is clearly made of something and it takes up space. If reality is subjective then what is the difference?