These rules are set limits that no one would ever think to oppose because they believe whole heartedly and why shouldn't they it is what everyone else believes. Morpheus and his crew however broke free from this system of force feed beliefs and because of this they are no longer limited by the matrix's basic rules. The most basic rule that is bent and later shattered is gravity. In the first scene of the movie Trinity is able to leap across rooftops that are extremely far apart because of her ability to deny what she had been conditioned to believe, she was capable of and to surpass her limitations and clear the gap. Trinity however can only bend the rules of the matrix because she cannot completely let go of her beliefs and so she cannot completely free her mind. Neo on the other hand never truly accepted the reality he was given, the life of Thomas Anderson. He lived his life as Neo in search of the truth because of a splinter in his mind that drove him to find Morpheus who would show him what he was never able to see. Since Neo could eventually totally free his mind he was able to not only bend to rules of gravity but he can completely deny the limits and fly.
Once the Neo comprehends that the matrix is a made up world he can free his mind of the system of belief that has trapped him and limited him to normalcy. In doing so he can see around all the "mirrors" and instead of seeing only a portion of reality he see the whole. He understands that only his mind is in the matrix not his body and a free mind is capable of anything. He is able to surpass not only the basic limits of the made up reality but he is able to manipulate the reality to match his perception. He understands that as Morpheus put it his strength and speed have nothing to do with his body within the matrix it is all based on the power of one's mind. Once he is able to see beyond his own body he can move faster and be stronger than he previously was.