I think this question has bothered a lot of people and they can appreciate having a sensible explanation to consider. I do not think the Matrix within a Matrix theory is a sensible explanation- only convenient.
I’m going to explain this by taking you backwards through the trilogy.
I can only recall two ‘powers’ that Neo possesses outside of the Matrix:
He fries the Sentinels
He has sight after he’s blinded.
It’s been a few years so maybe I’m overlooking something. Most people ask about the fried Sentinels. I’m going to talk about the sight first.
Bane blinds Neo in the third film. Neo is then able to perceive this red and amber representation of the world. It looks almost like heat vision but it doesn’t pick up standard heat signatures. It is something radiating, though. We know it isn’t as simple as heat because Neo cannot see Trinity. She’s been impaled and she’s lying beside him. He calls for her. He follows the sound of her voice fumbling around the debris before finding her body. They speak a few words and she tells him she’s been impaled. Neo is surprised. He doesn’t want to believe it. He feels around and discovers it’s true. Moments before that he was her copilot helping her navigate the expansive fields of pods.
Neo is directing Trinity to find the machine city. He surveys the fields of pods and a great machine metropolis in the distance before fixing his attention off in another direction. He points and says, “There. Those treetops.” But they aren’t treetops. They are power lines. Tremendous power lines. Trinity notices delicately correcting him. Neo ‘sees’ but I think it is more accurate to say he senses or perceives.
Near the end of the second film Neo and company are in their ship when the Sentinels group up outside of a particular range. One of them swirls around and hurls an object at the ship. Neo knows it’s a bomb. These Sentinels are a good distance away. The crew quickly exit the ship fleeing on foot over uneven terrain. It’s an impossible escape. Neo resolves, “we aren’t going to make it.” Trinity insists, “we have to try.” She scales an obstacle and turns to see Neo stationary. He says, “Something is different. I can feel them.” We understand what he means. He doesn’t feel them in a tactile way; he senses them. He turns extending his arm and the Sentinels gyrate wildly before dropping to the ground mere seconds before Neo collapses. He finds himself in limbo between the Source and the Matrix. He’s jumped from his physical body that was not jacked in to the Matrix and landed in the train station that programs use to travel between machine city and the Matrix. Physically, though, he’s lying in a med facility on a ship next to Bane.
During a conversation with the Oracle, she explains powers possessed by the One saying, “they extend beyond this world.”
NEO: Tell me how I separated my mind from my body without jacking in. Tell me how I stopped four sentinels by thinking it.
ORACLE: The power of the One extends beyond this world. It reaches from here all the way back to where it came from.
ORACLE: The Source. That’s what you felt when you touched those Sentinels.
It’s now the end of the first film. There’s a series of numbers on the screen as we’ve seen in the Matrix. The trace attempt halts and the screen flashes a warning – “carrier anomaly detected”. Neo is talking into the phone saying, “I know you’re out there. I can feel you now.” He’s talking to the programs operating within the Matrix. He’s broadcasting his message on a phone but he’s talking directly to the machines. Quite a long way from conversations about rabbit holes, which brings us nearly back to the beginning.
Morpheus offers Neo two pills, he explains that the red pill is part of a tracer program designed to disrupt input/output carrier signals. When Neo ingests the pill, it causes his carrier signal to drop the Matrix feed. A little worker bee Sentinel rushes over to assess the situation. Neo removes the tube from his throat and stomach just before the worker grips him revealing an array of sensors and scanners. He’s unplugged from the Matrix and flushed down the sewer slide. Once in the waste water, he’s promptly picked up by Morpheus et al.
You might suspect how Neo seems to have powers outside of the Matrix. The very important clue is the carrier signal. Consider that the Sentinels are in constant contact with the machine city. Yet they have no wires. The command and control infrastructure of the machines is wireless. It’s signals based. Hey, it’s a frequency on a spectrum. But what frequency? What spectrum? Doesn’t matter. What matters are these signals.
At this point, there’s enough to explain that Neo interacts with the Sentinels over a carrier signal they use to communicate with machine city. We can further explain that when Neo is blinded, he’s better able to sense or perceive the carrier signals emitted by the machines. That explains why he cannot see Trinity but he can see all of the machines. Additionally, he can see Smith in Bane’s body. That’s a bit different. The one common component there is the code. It’s a small step explaining that a carrier signal is not the actual message. It just carries it through a medium. The message is modulated input code. The modulation of a signal over the carrier signal allows the receiving station to interpret specific information. Ergo Neo ‘sees’ the modulated code being communicated over the carrier signal. That modulated code shares a base element with the program imprinted upon Neo. So can anyone from the Matrix do this? No.
When the Oracle states so plainly that the One has powers beyond this world (the Matrix) she’s revealing that the powers are linked to both the machines and the Source. Typically we think of the Source as the source code, which is partly right. It’s the source of the carrier signal upon which the the machines modulate code. It’s the broadcasting tower. The powers of the One extend to the Source because the power is the ability to modulate input code over the machines’s carrier signal(s). In the Matrix this means flying and stopping bullets.
Before things get too deep consider that we perceive radiation (heat) the same way Neo perceives these carrier signals. The reason no one else can sense the Sentinels and control them reaches further back than the movies. Only the Architect exposes any detail regarding the One’s physical difference, on top of his cognitive difference.
The theory is complicated because it is assembled from highly subtle and nuanced clues given when the Architect explains to Neo why he is ‘here’. Neo is born genetically unique. Never mind the function of the One for a moment. The genetic difference is subtle but in the environment that the machines have constructed for humans, this small difference has tremendous implications. Quite a leap of logic so let’s build a bridge.
The evidence is presented almost entirely in the conversation between Neo and the Architect. My explanation of this is a separate comment indicated in the footnotes. There are numerous minor points scattered throughout the trilogy but we have plenty in the exchange.
NEO: Why am I here?
ARCHITECT: Your life is the sum of a remainder of an unbalanced equation inherent to the programming of the matrix. You are the eventuality of an anomaly which, despite my sincerest efforts, I have been unable to eliminate from what is otherwise a harmony of mathematical precision. While it remains a burden assiduously avoided it is not unexpected, and thus not beyond a measure of control. Which has led you inexorably….here.
NEO: You haven’t answered my question.
ARCHITECT: Quite right. Interesting…that was quicker than the others.
Neo is a thumb drive. He’s carrying the prime program and some other bits of code. The curious bit is that the Architect tells Neo he been altered by the process though he’s still human. Altered. Has everyone been altered by the ‘process’? What makes Neo different?
No matter. He hasn’t answered the first question. To this the Architect explains that Neo is the sixth integral function. This is the metric the Architect uses to quantify instances of the Matrix. The emergence of a unique human born in the Matrix begins his count. The first was an anomaly. Unexpected. Neo, on the other hand, is a forgone conclusion up to the moment he chooses to save Trinity. What to make of these integrals and anomalies and remainders? It’s rather simple, really. Genetic mutation. Unpredictable, uncontrollable, spontaneous mutations at the tiniest level of humanity require the Matrix to refactored with each definitive step of human evolution. It is minor, mind you. But it happens. You didn’t think humans remained genetically static all those years, did you? This is what the Architect cannot account for and concordantly why he needs the One to return to the Source with his ‘code’, as well as the prime program. The One, as a function in the Matrix, is a construct of the Architect and the Oracle’s making. The person who expresses this genetic mutation allowing the prime program to be imprinted upon his cortex is fed the prophecy as a way of channeling him to the Source. It’s easy enough to spot from the outside looking in, I imagine.
Neo processes information much more efficiently than others. Think of people who pick up languages like a sponge. The Architect admits as much remarking how quickly Neo realized his question was not answered. You could say more quickly but that’s only partly correct. Directly is better. The fact is Neo is more sensitive to signals. In his body that means signals traveling up and down his spine. But the machines hacked right in to Neo’s body and started pumping code into his cortex. Neo’s body assimilated the Matrix. It imprinted upon Neo because Neo is different. We certainly know is that Neo is different as a human and as the One. Everyone says so at some point in some way.
So Neo is telepathic because of a genetic trait? No. Neo simply has a uniquely sensitive, direct nervous system. It sends and receives signals in a very special way. Recall Dozer feeding him all those capabilities – “He’s a machine.” Trinity bewildered in the first film – “How’d you do that? You move like they do.” Recall Neo fighting Smith at the end of the original when he’s moving in such a way that Smith appears in slow motion comparatively. See? Neo’s signals abilities are incredible. It just so happens that Sentinels have incredibly sensitive radios integrated into them. These radios can pick up a signal broadcast from the Earth’s surface by machine city even when they are miles beneath the surface near the Earth’s core, where it is warm.
Neo is becoming aware of this sensitivity he has for signals by the end of the first film. He feels the programs. Then he feels the Sentinels. When he dies, he’s discovered that losing his sight reduced the noise pollution in his nervous system (no more visual noise) and that freed him up to better register the carrier signal of the machines. Turns out his very sensitive nervous system was overwhelmed by so much sensation. Eliminating a sense made it easier to recognize how he was different. So the ending… The Architect and Oracle settle for 99.9% allowing the genetically unique and refusers to voluntarily be freed.