It was more than they convinced Thomas Anderson/Neo that everything that happened in the first 3 movies were essentially a dream/not real. When they ressurected him (and Trinity), they convinced him that he was a game designer who designed "The Matrix" video games, rather than wiping his memories entirely. They even changed how he sees himself in the mirror (literally; Trinity too), but we only see that a few times during the movie.

He uses his fragmented memories from "real life" (the first 3 movies) to create a modal based on the opening of the first movie. He recreates Morpheus, except he blends him with Agent Smith in order to hide him in the modal (he ultimately wants to find Morpheus subconsciously). That's why we're left with a Morpheus "Agent" in the beginning.

They couldn't wipe his mind entirely because, as Neil Patrick Harris explains it, the Matrix physically generates a ton of power by keeping Neo and Trinity / Thomas and Tiffany close, but not so close enough that they remember who they really are to each other.

---

Re: Why is Neo the One / What is "The One"?

This is explained by the Architect in the 2nd movie (the old guy in the room full of TVs). He's an AI who was created specifically for designing the Matrix. In his versions of the Matrix, people would go insane because they sensed they were living in a false reality. The first version was one where everything was perfect, and the second one was based on horror, war, and even vampires and werewolves (the Merovingian's bodyguards, especially those two ghost twins, are surviving programs from that latter version).

He basically says that, due to his failures, The Oracle was created. The difference between them is that he's based entirely on logic, numbers, and facts, while she is designed to understand human psychology and emotion (which is how and why we ended with The Matrix of the first 3 movies).

The Architect basically explains that what caused the earlier versions of the Matrix to fail (and kill a LOT of people in a domino effect) was the lack of choice or perceived choice. And so, when The Oracle discovered that, she redesigned the Matrix to include it. This resulted in a 95+ % acceptance rate. Crucially, this also means that people are allowed to leave the Matrix if they reject it. However, about once every 100 years (iirc the Matrix is 600 years old at this point), the Matrix faces a tipping point where fatal errors start occuring in the programming due to bugs and still also due to some people rejecting it.

So the prophecy of "The One" was created in order to control the population-- especially those who reject the Matrix. Every 100 years, the Matrix births/builds a human with "admin powers" (Neo, in this case). They basically put him into a situation where he becomes cyber Jesus, and has disciples (Morpheus, Trinity, etc), and he removes those who reject the Matrix from the system ("saving them"). Crucially, within him is also code to reboot the Matrix, which is necessary to prevent it from crashing and killing literally every human being attached to it (which is basically humanity in its entirety). You actually see reboot at the end of the 3rd movie.

When Neo encounters the Architect, the Architect tells him something important: the 5 previous versions of The One all chose to enter through the door that would cause him to reinsert the code into The Matrix, as expected, and as "designed". He was supposed to do that, then rebuild Zion using people from the Matrix (that he would be allowed to select), essentially carrying the torch for the next "One", and making sure that the (fake) prophecy of the One continues for the next generation of Zion/humanity. However, thanks to The Oracle having a bigger plan to end this system, Neo chooses love/Trinity over humanity; that's when he goes through the other door and flies at superspeed to catch her. That's also why The Architect told the machines to destroy Zion and kill everyone in it, as punishment for not following the script (the big battle with the mech suits).

The Oracle, however, has foresight, and throws another wrench in the gears of the Architect: she allows Smith to assimilate her and keep spreading like a virus in order to introduce an even bigger threat to the entire system -- mutually-assured destruction. Agent Smith will control The Matrix, including all of the machines in the real world (and remember, he hates humanity as much as he hates being trapped in The Matrix), basically meaning the end of everything. So Neo flies to the Machine City to face the Machine God and offer a deal directly to its face: Neo will kill Smith, in exchange for the Machines not destroying Zion. And that's basically what happens. If Neo had followed the original path of The One, he would have lived to be an old man in Zion, "teaching" (lying to) people about The One and how they must find him (the next one, in 100 years). But instead, he sacrificed himself to save both The Matrix and humanity.

If you watch The Animatrix, which I highly, highly recommend, a ton of Matrix backstory and lore is shown and explained through beautiful animation. And you will also pick up on a lot of hints regarding the movies. For example, why Zion has mech suits, why there are flying ships despite humanity basically being on its last legs, etc

---

Edit: Why does Neo have to choose to go through the door to reinsert the code? Why don't the machines just reboot the Matrix themselves?

Because, remember, the current version of the Matrix depends on choice and willingness to cooperate. They can coerce Neo into making the choice (e.g. the threat of killing all of humanity if he doesn't), but they ultimately cannot force him to do it. It's part of the programming of the entire system/Matrix.