If there’s one thing the MCU needs it’s more magic. Probably? I don’t know, you decide.
In an old library the sorcerer Kaecilius and his zealots string up the librarian and decapitate him before grabbing a forbidden tome and ripping out several of the pages. A hooded figure warns him that the ritual he has stolen will not give him what he wants but he calls her a hypocrite and flees through a portal to the streets of London. She follows and attempts to trap the zealots in a shifting twisted landscape of roads and buildings, but they are able to escape with the ritual.
Meanwhile, Doctor Steven Strange is in surgery, showing off his skills and kind of being an arrogant bastard. He saves a guys life and we are introduced to his on again off again ex, Christine. On the drive home he becomes distracted by a new possible case and his car careens off the side of a cliff. The accident nearly kills him, and causes severe nerve damage in his hands, effectively ending his career.
Strange spends months exhausting medical possibilities for recovery and is frustrated that he is unable to do anything, burning through his remaining funds and driving away everyone in his life, including Kristine. During physical rehabilitation he is told about a man that lost the use of his legs, fully paralyzed from the waist down, but somehow managed to find a cure. After his last options run dry he seeks that man out and is told that he had turned to mysticism, traveling to a place in Nepal called Kamar-Taj, and was healed. Seeing no other option, Strange uses the last of his money to travel to Nepal.
In Nepal, Strange is attacked by muggers, but is rescued by Mordo, who leads him to Kamar-Taj and introduces him to the Ancient One. They tell him that they are sorcerers, and that the paralyzed man was healed through magic, but he does not believe them, so they give him a demonstration. Surprised but convinced, Strange asks the Ancient One to teach him, but she refuses, and they kick him out. He spends two days asking for entry, and Mordo suggests to the Ancient One that a man of his skills could be useful, especially with the eminent threat that Kaecilius poses. She agrees and they allow him in to being training.
Strange is introduced to the concept of magic being energy pulled from other realms, spells, rituals, and relics. They teach him how to use a device called a sling ring to form portals to other locations, and as his prowess quickly advances, he is introduced to the mirror dimension, similar to our own, but with a very lose grasp on the laws of physics, where space can be folded, so he can practice in safety. He also meets Wong, the new librarian, and learns about Kaecilius and the Zealots. Meanwhile, Kaecilius implements the first part of the ritual, gaining power from the dark dimension, allowing him and his followers to become more powerful than ever.
In his zeal to learn all he can about magic, Strange sneaks access to a powerful relic: the Eye of Agamoto, and experiments with it to manipulate time itself, using it to temporarily restore the pages that Kaecilius stole from the book. He is caught by Wong and Mordo and heavily admonished, warned that messing with things he does not understand can have dire consequences, and that he should not disrupt the natural order. Upset by the rules and restrictions, he demands to know what it is that they do, and they tell him that the sorcerers protect the Earth from threats from other dimensions, especially Dormammu, who rules over the dark dimension where time does not exist, and who wants to destroy Earth most of all. They tell him that Dormammu is held at bay by the three sanctums built by Agamoto in London, Hong Kong, and New York. He realizes that the ritual Kaecilius stole was to summon Dormammu, and decides that he wants out, all of this being far too dangerous and thankless of a job for him, but before he can do anything else a portal opens, and a sorcerer from the London Sanctum tells them that it is under attack by Kaecilius. A blast of fire as the sanctum is destroyed throws Strange through the portal to the New York sanctum.
Stunned, Strange hears a commotion downstairs and finds that Kaecilius has moved on to the New York sanctum. He watches as the sanctum’s master is killed, and finds himself standing up to stop Kaecilius himself, as there is no one else to do so. He manages to fight off a few of the zealots, trapping them or sending them through portals far away, and in the process picks up the cloak of levitation. He is able to trap Kaecilius in magical bonds, and Kaecilius tells him that he should be helping him. He says that his mission is to save the world from death, and that Dormammu offers eternal life for everyone. He also warns strange that the Ancient One has been lying to them, that her unnaturally long life is due to her drawing power from the dark dimension.
Before he can say more, Strange is attacked by another of the zealots and is stabbed. The cloak wraps itself around the zealot’s head, holding him down, and Strange quickly opens a portal to his old hospital, seeking out Kristine for medical attention. As he lays dying on the table he steps out of his body and into the astral plane, offering her advice on how to help him, but he also finds that the zealot has done the same and followed him. He fights off the zealot as Kristine fights to save his life, and he uses the electrical jolt from the defibrillator to kill the zealot. He apologizes to her for his behavior in the past and returns to the sanctum to meet Mordo and the Ancient One, but finds that Kaecilius has escaped his bonds. They argue about what Kaecilius told Strange about the Ancient One, but are soon interrupted by Kaecilius’ return. Having taken their sling rings in the earlier, Strange attempts to trap the zealots in the mirror dimension, reasoning that they will be unable to escape, but Mordo points out that with power from the dark dimension they will be incredibly powerful there, and they find themselves running for their lives in the twisting New York landscape as the zealots attempt to steal their rings back. The Ancient One catches up to face Kaecilius, and during the battle confirms that she is drawing power from the dark dimension, before suffering a long fall into the pavement on their escape.
Strange and Mordo rush the Ancient One to the hospital, but nothing can be done, and in her final moments she imparts some wisdom to Strange in the astral plane. After two failed attempts on the New York sanctum, Strange and Mordo assume that Kaecilius will next attack the Hong Kong sanctum, and rush there only to find that the sanctum has already fallen, and that Dormammu is nearly able to reach Earth, with the dark dimension clearly visible in the sky over Hong Kong. Still wearing the Eye of Agamoto from his earlier experiment, Strange turns back time, undoing the devestation to the area and bringing a defeated Wong back to life, but he is stopped before he can fully restore the sanctum as Kaecilius and the zealots break free of the spell and approach them. Forming a new plan, Strange launches himself into the dark dimension to face Dormammu directly.
Approaching Dormammu and asking for a deal, Strange is quickly skewered and killed, but only a moment later appears again. He explains that he brought time into the dimension with him, specifically a time loop. Over and over again Dormammu kills him, in a variety of ways, but always he comes back, and Dormammu finds it infuriating. Finally ready to bargain, Dormammu agrees to take the zealots with him into the dark dimension and leave the Earth alone for good if Strange will agree to break the time loop. Strange returns to the moment he left, the zealots are dragged into the dark dimension, and Strange completes the restoration of the sanctum.
Despite their victory, and still angry over the Ancient One’s betrayal, Mordo is livid that Strange broke the laws of nature to achieve win, claiming that there will be consequences, and breaks ties with the sorcerers of Kamar’Taj and Strange returns to the New York sanctum to serve as its new master. Months later, Mordo visits the man that sent Strange to Kamar’Taj in the first place and steals his magic, claiming that the problem with the world is that there are too many sorcerers.
After the credits we see Strange interview Thor, offering to help him and Loki find Odin in the hopes that it will get Loki out of New York faster.
TLDR: Talented Doctor Steven Strange finds his life in ruins after a devastating car accident and seeks help from the sorcerers in Kamar’Taj. He quickly learns the magical arts and assists in stopping a rogue sorcerer that wants to give the Earth over to an evil being called Dormammu, but in the process creates new enemies.