Why does Trinity's kiss revive Neo in The Matrix?

First, a correction of a misconception. The Oracle never tells Neo he's not the one, he tells himself that. The quote from the Oracle/Neo is:

Oracle: Sorry, kid. You got the gift, but it looks like you're waiting for something.

Neo: What?

Oracle: Your next life, maybe. Who knows? That's the way these things go.

The sorry is a little ambiguous, and could easily be taken to mean "Sorry, you're wrong on that". However, the last part of the conversation is the important part.

When Neo gets "killed" in the Matrix, his body is still around, and consciousness presumably hasn't departed yet. Trinity confesses that she was destined to fall in love with the One, and she loves Neo, so he couldn't be dead.

This resonates with the consciousness that is still there, and he cognitively realizes and dissociates himself with the concept of death being tied to the Matrix, and is "reborn" in the Matrix. This fulfills all parts of the prophecy, Neo is on his "next life", Trinity is in love with him, and Matrix go boom.

An explanation:

How about this explanation from thematrix101.com:

Q: How does Neo come back to life? Is it Trinity's kiss?

A: Not exactly, though interpretations differ slightly. Simplest explanations are always best: Neo is the One. Trinity was told by the Oracle she would fall in love with the One. The One will save humanity and help overthrow the machines. Once Trinity voices her love to Neo, it's like the last link in the prophecy chain - he must be the One, so he can't die until he's done what the One is there to do. Once that link clicks into place, it's apparent that Neo can do whatever he wants in the Matrix.

It seems reasonable enough - her kiss is Neo's way of realising he is the one. Hence his choice immediately after to fight Agent Smith instead of running as he's been told.

A justification based on the script

To back this up a bit more, I paraphrase the following from the script of the scene at http://dc-mrg.english.ucsb.edu:

Neo gets into the hotel, and runs along the corrider until he reaches room 303. He can hear the sound of the phone ringing from within the room, as he opens the door. He comes face to face with Agent Smith, who looks him straight in the eyes, and shoots him. Neo touches his wound, as blood comes out onto his hand. He looks up at Agent Smith, as he shoots Neo again, and he stumbles back against the wall. Agent Smith shoots him a few more times just to make sure, and he slides down the wall, and slumps on the floor.

Trinity and Morpheus watch his life signs go dead.

...

Agent Smith : Goodbye......Mr. Anderson...

Trinity is still sitting by Neo.

Trinity : Neo..I'm not afraid anymore! The Oracle told me I would fall in love, and that man, the man that I loved, would be The One. So, you see, you can't be dead...you can't be... because I love you....you hear me? I love you...

She leans over and kisses him

As she says this, the blow echoes deep in Neo's mind. In the Matrix, his eyes snap open. Trinity screams as his life monitors jump back to life. Tank and Morpheus stare, in disbelief.

Trinity : Now get up!!

Neo sits up, trying to focus. He stands up, and the Agents turn around, staring at him, also in disbelief. Agent Smith grimaces and aims his rifle at Neo. He fires.

Neo : ...No...!

As Agent Smith fires, Neo just puts his hand out. The bullets slow down and stop in front of him. Neo looks at them, and picks one up. He looks at it, and then drops it to the floor. He looks at the rest, and moves his head down. They all fall to the ground. The Agents stare, and all take a step back.

Tank : How!?!

Morpheus : He is... The One...

Remember the matrix is a program, and the choosen one had several requirements, one of them was that he did must believe it, and another one was that trinity should fall inlove of him. So it triggered choosenOne = true, and it started the process...

I believe Trinity's kiss revives Neo not because of the physical kiss itself but because love does "its thing". It sets a 'connection' that was missing, makes Neo understand the new world in a light he intimately knows, which machines are unable to grasp, he understands what makes humans human and somehow that completes a circle that no machine nor program nor corrupt system has been able to complete, and powers him up.

Let me explain.

Machines or programs cannot love. They just think of love as an illness of sort.

Same happens to institutions. They just rule, set and execute laws, always looking for self-preservation.

Love is totally different. There's love as emotion, of course, which is limited, and there's a different kind of love that Neo and Trinity feel, share and even identify themselves with.

Love can even be self-deprecating. A machine or program will never understand that. They may replicate emotions but cannot replicate true love. Love's main difference is that it's not based on power over others.

Neo's life at the beginning is that of a rebel, hacking all over and beyond. To be a rebel is to be a lover deep down, to fight for some values you love beyond every stablished rule (rules as truths as opposed to Truth).

Up to the kiss scene, Neo has had no time to love again. He's been so busy rediscovering the world and trying to master the new 'laws' (understanding the world as he now realizes it is, learning awesome skills, re-learning who he may be...). In a way, learning (at least how Neo does) is power-based. He needs to master each new skill, fully focused. Focus is a key word, because while focus (and power) distinguishes one thing from another, set limits and define by the differences, love integrates.

Back to the point. Power is not all there is. The kiss connects him to his true nature, to the old Neo, and helps him understand this is not a new world but the same, he has not to be a new different one but who he has always been, just see in a new light.

He has always been a fighter for love, and he still is. Before there were the institutions and he knew their weak points, and his drive took him to learn how to stop their abuse over people. Now it's not that much different in essence, just more difficult not to be deceived. He knows what they are doing (which he also masters) and he knows what they can't do and never will. What he was missing was their weak point, and Trinity's love enlightened him.

"Being the one, is just like being in love. No one can tell you you're in love, you just know it. Through and through. Balls to bones."

The Oracle never does anything supernatural. Its predictions could as well be based on what will be the only reasonable choice and information obtained in normal/program means.

No one knows you are in love before you know it, atleast in a subconscious way, you get nervous or behave differently in your love's presence, at which point someone else could tell from the signs. So her analogy may be exactly true: Unless you know you are the one, even if only subconsciously, she can't tell if you are or not. This would explain her "tests", like "open your" and what she says afterwards and also the sentence "Who knows? That's the way these things go.".

So the reason for her saying that he has a gift may as well be, that he is the only one who was in doubt, when asked whether he was the one or not, while others rejected right away.

So now to his death. There are 2 deaths in medicine, heart dead and brain dead. If your heart dies your body's oxygen supply is cut off. After all oxygen is used up, your cells start dying, including those in your brain. This is why people can be revived after a heart death, by pressing the chest in a rhythmic way or by electric currents, but not after being shot in the heart or head.

Agent smith never shot Neo in the head. The first and all subsequent shots went into his body. That means that his brain was fully functional, for atleast 1 minute after his death. This is also the reason why he can revive Trinity.

We also don't know where exactly the life signs are read from, because they show a 3D picture of his heart with a lot of text data on the sides. So maybe the image is obtained from the matrix data and doesn't necessarily show Neo's real heart.

Now to the kiss. From the deception earlier in the movie, where the rogue mad guy killed half the crew, we know that doing something in the real world will have the same effect on people in the matrix that are hooked up. So the kiss from Trinity will have given him the feeling on his lips of being kissed, while he may not have been able to hear her words at all. This sensation may have shocked him and let him realise that he was not in fact dead and his real body was still intact, although his mind only could have been damaged, because of the mind link between real world and matrix. This realisation may have been enough to let his brain give the right signals and let his real heart pump again, which leads to his revival in the matrix (as well as in the real world). This rise from the dead may have given him the confidence boost to believe, that he was the one, which may be all that is necessary to unleash those powers.

After all the oracle says in Matrix Reloaded, that Neo has turned her into a believer and in the first part she never mentions his revival in a convincing matter in her prophecy. She only says, "Your next life, maybe.", which could really mean anything. So his revival may have been a surprise for her after all.

When Neo is in the elevator with trinity he says that the spoon is not real which was told to him by the kid before meeting the Oracle. They are also told that if the mind does in the matrix than so does the body. That being said when he gets shot by agent Smith at the end of the first matrix he touches the bullet wound at then falls back shortly after. At this point he is still alive and then trinity shortly after says that she loves him and than he gets up as if nothing happened. Well, if you think back to when he says the spoon isn’t real than the bullet isn’t real either and I think he realizes this and the body brain doesn’t die and therefor neither does he. Than trinity tells him I love you and that your the one and bam.

Not sure why the answers are omitting the part that I think makes the 'revivving of Neo' have particular meaning rather than being weird.

https://matrix.fandom.com/wiki/Thread:7715

In the first Matrix film, after Neo saved Trinity from the helicopter crash, the pair were in the subway station waiting for the phone to ring so Trinity could jack-out. Just before Agent Smith appears, Trinity says to Neo: "Neo, I want to tell you something, but I'm afraid of what it will mean if I do. Everything the Oracle told me has come true...everything but this." She hesitates, and then Smith shows up to kill them and interrupts her.

So Trinity is in dilemmaville throughout the story... until near the end when she is standing over a "dead" Neo. It is at this moment she gets it. In an act of faith --and a nice twist on the `Princess kisses the frog' fable-- she kisses Neo and he awakens. She verbalizes her faith just before kissing him with the line: "Neo, I'm not afraid anymore. The Oracle told me that I would fall in love with a dead man. And that man, the man that I loved would be the One. So you see, you can't be dead. You can't be, because I love you. You hear me? I love you."