Neo must obviously have powers beyond the matrix because he knew that the squids where throwing a bomb, he destroyed the squidies with his "powers," and he has visions of the future in his dreams. To get a better grasp on what neo is doing you have to take a look at thing metaphysicaly.
Reality is the illusion that everything we encounter exists as a separate and distinct "thing" from our own existence. This is an arbitrary distintion made up by our physcy. Nothing can be proven to be real under this thinking.
"all that exists as persons, places and things lives only within our own consciousness" - goldsmith
I believe Neo power comes from his awakening to this truth.
"It is only because of ignorance that the Self appears to be finite. When ignorance is destroyed, the Self, which does not admit of any multiplicity whatsoever, truly reveals Itself by Itself, like the sun when the cloud is removed." - sankaracharya
under this thinking everything is really one, and neo as the annomly has transended our reality and become "the one." thusly he can can effect all others because they are an extention of "himself" and he is an extention of them. In the end this only my interpretation but i think that this is the line of thinking that the brothers go along, damn those guys are smart.
Another point I find interesting is Agent Smith. I think that when neo attacked agent smith at the end of the first movie the thing he "inserted" into smith was his ego, his sense of himself that chained him to this interpretation of reality. This frees neo from the illusion but also traps smith with in it, Smith has become the super-ego while neo has become the id. Smith is driven to repress and control all while neo is driven by the pleasure-principle(ie. instant gratification). I can't wait to see what happens in their final battle, Ego vs. Id very exciting.
Oh, and sorry for writing so much 😗