The full story
Loki starts where Avengers: Endgame's trip to 2012 left him: grabbing the Tesseract in the confusion after the Battle of New York and vanishing. He lands in the Gobi Desert and is promptly arrested by the Time Variance Authority, a bureaucracy outside time that polices the Sacred Timeline, the one history the Time-Keepers have ordained. Anything that strays from it is a "variant," and variants get pruned.
Judge Ravonna Renslayer is ready to prune him. Agent Mobius M. Mobius, who hunts variants for a living, has a better use for him. He sits Loki down in the Time Theater and shows him the life he was meant to have: the part he plays in his mother Frigga's death, and Thanos killing him on the Asgardian ship. Loki tries to escape and finds Infinity Stones being used as paperweights in a TVA desk drawer. He agrees to help Mobius hunt a variant of himself who has been killing TVA soldiers and stealing their reset charges.
The trail runs through a Renaissance fair in 1985 Wisconsin, the eruption of Pompeii and a 2050 Alabama hurricane: the Variant hides inside apocalypses, where nothing she does can change the timeline. In Alabama she takes over Hunter B-15's mind and turns out to be a woman, Sylvie, who sets off the stolen charges across history to branch the timeline in dozens of places at once. Loki follows her into the TVA and pulls them both to Lamentis-1 in 2077, a moon about to be destroyed. Stranded with a dead TemPad, they start to trust each other, and Sylvie tells him what she's learned: everyone who works at the TVA is a variant, with their memories wiped.
The bond between two Lokis is a Nexus event big enough for the TVA to find them. Mobius finds proof that Renslayer covered up what Hunter C-20 said after Sylvie captured her, that the TVA's workers are variants, and Renslayer has him pruned. B-15, whose memories Sylvie has restored, turns on Renslayer. Brought before the Time-Keepers, Sylvie beheads one, and they turn out to be androids. Renslayer prunes Loki.
Pruning doesn't kill. It dumps variants in the Void at the end of time, guarded by the cloud-like monster Alioth. Loki meets a crowd of his own variants, among them Classic Loki, Kid Loki and an alligator. Sylvie prunes herself to follow him, and finds Mobius there too, pruned before him; he takes the TemPad she brought back to the TVA. Classic Loki dies buying them time, and together Loki and Sylvie enchant Alioth, which opens the way to a citadel beyond it.
Inside is He Who Remains, the TVA's real creator. A scientist from the 31st century, he discovered other universes; his variants among them waged a war across the multiverse until he used Alioth to end it, built the TVA, and pruned every branch since. He's tired, and he offers them a choice: take his place, or kill him and let his variants return, worse than before. Loki hesitates. Sylvie kisses him, sends him back to the TVA, and kills He Who Remains, and the timeline branches without end. At the TVA, Mobius and B-15 don't know who Loki is, and the statues of the Time-Keepers have been replaced by one of He Who Remains' variants.