King of Monsters focuses primarily on the Russells, a family that was caught in the rampage that made up the finale of Gareth Edwards' earlier movie, in which Godzilla battled some Mutos and saved San Francisco, but created a big mess in the process. The stomping beasts killed Mark and Emma Russell's son, and in their grief the couple split. She (Vera Farmiga) went to live on a Monarch research base with their daughter Madison (Stranger Things' Millie Bobby Brown). Emma has been building a sonic device that can control and subdue the monsters, called Titans. Meanwhile Mark (Kyle Chandler) has dedicated his life to photographing wolves. He's eventually brought back into the fold when Emma and Madison are kidnapped by eco-terrorists led by a scowling leader named Jonah Alan (played by Tywin Lannister, a.k.a. Charles Dance).
Except -- in a headache-inducing twist -- that was part of Emma's plan. She's teaming with Jonah against Monarch because she misguidedly thinks she can save the planet by letting the Titans roam free, thinning humanity and its pollution. Like Thanos before her, Emma's ostensibly noble intentions cause a litany of issues issues when it turns out that one of the ancient giants that Monarch has been monitoring, the three-headed King Ghidorah, is especially cruel and powerful, and not willing to submit to Godzilla's more benevolent impulses.
If this sounds needlessly confusing, well, it gets worse when you factor in that the script seems cobbled together from various unfinished drafts. It casually drops references to characters and scenarios in a way that assumes the audience has a bigger investment in the narrative the film ever earns. Perhaps the most baffling moment is when it's revealed, with little to no explanation, that Ziyi Zhang plays identical scientists on different sides of the world who may or may not be twins.