»Neo's Power Explained«

Neo has no power. He just wakes up.

His power stems from his becoming conscious of his surroundings. At first it was the realization that the world of the Matrix was a lie. He became aware of it, its sets of rules, and thus by exerting himself against it, he was able to seem superhuman, when in reality he was the same person, he just rejected the fake identity the Matrix told him he was supposed to be, and thus "empowered" himself.

In the Architect's room he is told that he is still playing a role assigned to him, that he has a destiny to fulfill, a fate he cannot avoid as the One. But Neo rejects this too. Because of that, he gains the power of the source. The choice was to either bow down before the source and obey your function, keep the Matrix going, OR... not. He gained access to the source when he entered the room! That was the point of the keymaker. He was there. It was in his hands what to do with the power behind the Matrix, and he refuses to obey it.

He doesnt become superhuman- he just inherits the fortune that was meant to be his all along. He is the king of the kingdom and they were trying to trick him into signing it all away to the next one. Because he didnt he has total authority over the matrix- his power therefore exceeds it borders, stemming directly from the source.

Just like he didnt fully realize what he was capable of in the first movie, becoming gradually more and more potent as he believes in himself, neo doesnt fully grasp what he can tap into with this new consciousness of the Source outside the Matrix he now has authority over. His mind expands to the point that his body jacks out while his mind is still inside! Even when his mind returns to the body when it gets jacked back in (after trinity finds him) a part of his consciousness is still tied in with the source. He doesnt realize it until he loses his physical sight, but he can sense his power still in the machine world.

Is that too meta-physical for you? Not scientific enough? Sorry for your lack of imagination, but I think its pretty cool.

At last someone who sees something.
I explained this in "zion is real" but with less details indeed.
I would add that this is one of the reason The Wacho bro showed the MOBIL AVE (limbo avenue) sequence where we see neo in a matrix state similar to his real coma.