Owlboy's story is told without dialogue. Otus is mute, the cutscenes are animation and music, and the plot is carried by item pickups and environmental storytelling. Here is the shape of it.
The world of Owlboy had its landmasses lifted into the sky by an ancient device called the Anti-Hex, leaving an empty ocean underneath. The islands float in loose regions - Vellie, Tropos, Mesosphere, the Floating Continent, the Eternal Sanctuary, the Owl Temple and others - and travel between them is the core of the exploration. When the Anti-Hex is finally switched on at the end, the landmasses fall back into the sea. That image is the last thing the game asks you to sit with.
Otus lives in Vellie, a floating village of owls, and is looked down on because he cannot fly. His mentor Asio gives him the owl cloak - his wings - which unlocks the game's central traversal verb. From that point Otus is a genuinely capable creature, which is what makes the tragedy of the ending land.
Sky pirates led by Captain Molstrom attack Vellie and steal one of three ancient Owl relics - the Artifact of Doom - eventually destroying the capital city of Advent. Retrieving the relics and stopping the pirates is the spine of the campaign, and the dual-boss fight against Molstrom and his partner is the game's climactic encounter.
The true antagonist is Solus, a former owl who secretly manipulates the pirates to gather the relics so he can activate the Anti-Hex himself. Every pirate attack in the second half is really a Solus objective. The ending is deliberately bittersweet: the Anti-Hex is forged, Molstrom is destroyed, the landmasses fall back to the sea - and many die, Advent stays ruined, and Otus's own fate is left ambiguous as he falls toward the Earth.
Otus carries a rotating cast, and because he cannot use weapons himself, who you carry is a genuine tactical decision:
You can recall a carried companion with R, and you can throw companions to stun or reposition enemies. That one control is the mechanical heart of the game.