It connects to the way they can bend reality and physics. There is no spoon, means literally the spoon is a virtual object, but even more important is, when the boy goes on to say:
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, that it is not the spoon that bends, it is only yourself.
So, let's say the character was in a videogame. For the character, the videogame is reality, but the player himself, knows it is a simulation. A smart player can try to bypass that simulation by for example changing the internal clock of the machine to simulate the time change, instead of waiting 10 hours for an event, cheating the system. We see this often with online cheaters. Players flying around when they shouldn't, or unable to be damaged.
In the Matrix, the same person is the game character (his matrix avatar), the player (his real conscience) and the machine (his brain connected to the machine). Bending himself is literally, sending the unexpected data to the matrix, stopping following the rules the Matrix tries to impose on him. He is choosing for his machine (his brain) to send weird signals to the Matrix, making him able to for example fly.
Why does this happen? Why does the simulation allow it at all, instead of blocking "flying behaviour" for example? There are 2 kinds of networks in games. The first is a server - client system. In this way, the server is the one that decides what happens, and all the clients follow suit. If a client is misbehaving (doing something impossible by the game standards), his actions are adjusted to be "normal". The second is p2p systems (peer to peer), where every client share the load, and everyone trusts everyone else signals. Most games use p2p (because it's cheaper), and this leads to blatant cheating.
Why would the Matrix be p2p, instead of server - client? The brains are connected in p2p, and possibly host the Matrix themselves. The answer lies in a older version of the script (and the canon webcomic Goliath). In the movie release Humans are said by Morpheus to be used as an energy source - but in an early script, humans were used as CPUs. Their brains were networked in a peer to peer network to use as a super powerful computer and possibly also running the Matrix themselves. And very much like a p2p data sent by another "peer" is just assumed to be true by the system.
So, by realizing the spoon isn't real, realizing he is a peer in the network, and bending himself by sending unexpected data, he can make himself fly, stop bullets, and generally fool the system.