When neo finally gets there, unscathed, he encounters the architect, who mesmerizes him with what reviewers call 'mumbo-jumbo' because they are too stupid to understand what is going on. The architect knew which door neo was going to choose, but neo chose it slightly differently than the architect anticipated. He came to a realization right before he started walking towards the door. He realized why he made the choice. Upon this epiphany, he is able to 'see past it' as the oracle put it, and can now feel the machines, and shuts down a few sentinels. However, his ability to manipulate them comes from a link with the source. Upon opening that link, neo is sort of sucked into the machine mainframe, as he doesn't understand his own powers. The merovingian, quick on the draw places him under 'house arrest'.
Now I know, at this point, you are like "wait wait, what the heck, how'd he do it?". You see, humans have been living as crops by giving off bio-electrical energy for several hundred years. This process has changed humanity as a whole. They are sensitive to the electromagnetic fields given off by electrical machinery. They are also capable of exerting influence on these electromagnetic fields. If you want to call it a wireless connection, go ahead. It isn't exactly precisely that, but that's really close enough to what is going on. The oracle says that neo's power goes from the matrix to the source, and that neo touched the source when he stopped the sentinels, not before. Neo exerted his bioelectrics to stop them, coming into contact with the wireless conduit the sentinels have with the source, and well you know the rest now. We also know that something akin to a wireless connection must be the answer because of what neo demonstrates as powers in the real world. He cannot heal trinity, he cannot heal his eyes, he cannot move things with his mind. He can, however, destroy sentinels and sense the machines. His powers are limited to influencing and sensing the machines. He can also sense bane, because his bio-magnetic field now reads like the machines due to the infection by Smith.
Neo's ability to sense and manipulate the machines in close quarters fits perfectly with the bio-electricity theory. It also fits with the reading material keanu reeves had to do. Remember that he had to read a book about how humans were becoming more like machines. His ability to manipulate this electromagnetic field in this way is a representation of that transformation.
Now, neo predicting the future is a whole nother can of worms. The oracle tells him that he's already made a choice, and to see past it, he must understand it. Basically, she's telling neo that the world is deterministic. That future events can be predicted. Neo has 'the sight'. Apparently his consciousness has been changed, and he can see the future. I don't know how exactly this can be explained, but I believe that either the oracle implanted the events into his mind, and then let his mind work out whether or not trinity dies, or it's a sort of mysticism or philosophical belief the WB's are positing.
Unfortunately, there's not much more about this 'sight' in m3 to give us any sort of answers. However, smith, who's been copying himself all over, has now become a real threat to the machines, and this is not what the oracle planned. However, neo knows what he must do, he goes to 01 and stops smith. Peace love and harmony, but not quite. The architect is willing to free those who want out of the matrix. It's fine with him anyways since they just destabalize the place. He pretty much thinks that he doesn't have to worry about any more anomalies.
Why is smith so much stronger? Well, he seemed to gain a lot of power when he absorbed the Oracle. Neo basically knows that he cannot beat smith, not in a straight contest. So he loses. For a third and final time, neo does NOT win a decisive battle against smith, yet fights all the same. Some people are frustrated by this, and I can understand it, but that's just how it goes when you have a nemesis. Also, consider what the WBs may be trying to say about war and fighting. Perhaps they believe that fighting really does get us no where. Perhaps they believe you should fight even when you know you won't win. Maybe both... in a strange sort of way. But like I said, I'm not here to start deconstructing the psychological, political, spiritual, or religious connotations of the movies.
What's up with Sati, the little kid? She is the first program created out of love, and without a purpose. Smith says that without purpose they would not exist in m2, so here's what happens. Not having a purpose denies a program from a choice when it faces deletion. If a program has a purpose, it can choose exile. Sati can not choose exile, so her father makes a deal with merovingian to smuggle Sati into the matrix. Sati seems like she may have great powers in the matrix, and I believe this is due to her not being confined by purpose. Every other program is limited in their abilities based on what their purpose is. They can only do what is necessary to achieving that purpose. Sati represents a new kind of program, one that the oracle believes is 'higher' in a way than the current programs, because love is important. Love and freedom are valuable.
Some people also wonder what exactly happened during the super brawl, and wonder how exactly neo beat smith. See, smith is a program, we all knew that. And each smith is connected to each other smith. We already knew that also. Smith's purpose is to destroy neo. At least that is what he had made his purpose after m1. At the end of the superbrawl, smith thinks he has destroyed neo. What has actually happened is that smith is now connected with the source. However, by destroying Neo, he has lost his purpose. Then the source knows to strike. See, now that he's lost his purpose, the machines can delete him without giving the option of exile, and all the smiths go boom.
At the end of the movie, Sati makes a sunrise for neo, and I believe it is actually with the intent that he will see it. Perhaps even she has something to do with his resurrection. However, the Oracle also says we will see him again, and I believe that she doesn't just mean we'll see another anomaly. Another anomaly would not be Neo, contrary to some misconceptions about the movie. We do see Neo after revolutions, in the short clip at the end of the first Matrix. Neo then goes to the phone, and tells the machine consciousness that he is going to be making some changes. That the matrix will no longer be a prison, or enslavement. That it will instead become a place where people can live real meaningful lives by removing controls. Yes, that's right, the phone call at the end of m1 is out of order with the rest of the movies. You can tell this by reading the dates on the phone call trace program. It's 18 months from the phone call made by Trinity during m1, and m2 happens six months after m1. Pretty concrete evidence.
Some people think that the Oracle was just playing some sort of game with the Architect. That's not true, that's just how the program Architect interprets their struggle. To him everything is numbers and math and so he plays it like a game, like the Merovingian does. It's kinda like the WBs are telling men to grow up in these movies. In any case, the Oracle had planned everything out, yes, but it was not a game. She could see the future, she knew the possibilities, and she knew that only way out of the war was with peace.