The MCU has had a banner 2021 with more releases across more platforms and mediums than ever before and it's all coming to a head with Spider-Man: No Way Home, the third movie in the Spider-Man arm of the Sony/Marvel partnership and, arguably, the most highly anticipated Marvel movie of Phase 4 thus far. And for good reason--with the multiverse confirmed and the participation of vintage Spider-Man villains like Willem Dafoe's Green Goblin as a result, the buzz around the movie has been astronomical.
So, what actually happens when Peter Parker comes face-to-face with the bad guys of old franchises? We'll break it down. Naturally, tons of spoilers from here on out so tread with caution.
When Peter accidentally messed up Doctor Strange's spell--an attempt to make the world forget that Peter Parker was Spider-Man, he inadvertently pulled in people from other realities who did know Peter Parker was Spider-Man. This meant the villains that we've seen in the promotional material--Lizard, Electro, Sandman, Doc Ock, and Green Goblin were all abruptly pulled into the MCU to cause havoc in their confusion. Doctor Strange had a relatively easy fix for this issue, but Peter (and the villains) quickly realized that they were all pulled from their respective worlds moments before their deaths, and sending them back without changing something would functionally doom them all to die. This prompted Peter to hatch a plan to "cure" each villain before sending them back--which, naturally, wasn't easy to do.
Thankfully, the villains weren't the only ones pulled over. Other Peter Parkers--namely Tobey Macguire and Andrew Garfield--were also dimension-hopped into the MCU to assist. Three Spider-Men are, of course, better than one--but the entire situation was tempered by extreme tragedy. Not moments before the other two Peters were discovered, our Peter's Aunt May was killed in an attack by Green Goblin, just after giving the classic "with great power," speech. Only through a serious heart-to-heart with his alternate selves is our Peter able to pick himself up and walk back his grief and regret.
The movie then culminates in a massive fight between five villains and three Spider-Men that, thankfully, results in all five baddies being cured--but not before space-time itself begins to feel the strain (thanks largely to some chaos created by Green Goblin directed at Doctor Strange, who was vocally in the "there's no need to cure them, send them back home and forget about it" camp.) The spell, having officially gone completely out of control, threatens to bring in an infinite number of people who know Peter Parker is Spider-Man from an infinite number of realities--some of which even start to show up as silhouettes in cracks in the sky (the most clearly visible is an outline of Kraven the Hunter).
In a last ditch effort to prevent all reality from tearing itself apart (and to send all the now-cured visitors home for good) Peter agrees on a Hail Mary play with Strange. A new spell that would functionally make everyone across the multiverse forget Peter Parker ever existed at all. This would mean everyone--including MJ and Ned--would have no memory of ever being friends with Peter, let alone that he's also a superhero. It's never clarified how this will affect the other Peters in their own respective realities.
The spell is a success, everyone is set home, and Peter is left to pick up the pieces in complete anonymity. Prior to the spell, he promised to go find MJ and Ned and explain everything to them after the fact, but when the dust finally settles he can't bring himself to do it. Instead, he goes to the diner where MJ works, sees her catching up with Ned, orders a cup of coffee as a stranger, and leaves. He then gets an apartment--one that looks stunningly similar to the dump Tobey Macguire used to live in back in the original trilogy, and officially decides to continue on as Spider-Man (who no one has forgotten about) in a new, low tech, homemade suit.
While this clearly isn't the end of Peter Parker in the MCU, it will certainly impact his future. As it stands, he no longer has access to any of his Stark Tech, he has no family or friends, and his Avengers colleagues only know of him as Spider-Man. His future MCU appearances have yet to be confirmed, so it's difficult to speculate how this will manifest practically in any upcoming stories.
Being an MCU movie, there are naturally post-credits scenes to get into as well. The first deals directly with the post-credits scene of Venom: Let There Be Carnage, which seemed to pull Tom Hardy as Eddie Brock into the MCU--noted by J. Jonah Jameson on the TV after the magic dissipated. Apparently Brock found his way to Mexico where he spent some time at a bar, getting the state of the world explained to him by a bartender. Thanos, the Avengers, and of course Spider-Man were all covered topics, and Brock even had a momentary burst of inspiration to go track down Spider-Man in New York, but before he could actually act on this plan, the spell was reversed and Brock was (apparently) sent back to his home dimension.
There was, however, a tiny trace of the Venom symbiote left behind on the bar, implying that the MCU will be seeing a version of Venom in the future--though it probably won't be Hardy's take on the character.
It's also never made clear how the spell affected Brock in the first place, since it was only supposed to affect people who knew Peter Parker in their respective worlds, which Brock most assuredly did not.
Regardless, the tiny speck of symbiote left behind is definitely going to be an issue at some point.
The second post-credits scene was less a "scene" and more a teaser trailer for the upcoming Doctor Strange In The Multiverse Of Madness where brief snips of the movie were shared with little to no context. It's clear that Strange's repeated tampering with space time is going to be causing massive problems, he's going to go for Wanda for help, and he's eventually going to have to fight off a dark (possibly mirror-dimension) version of himself. It's likely that these clips and scenes will all be released widely at a later date as an official teaser for the movie.