In America, a boot is a type of footwear. In the UK, the boot refers to the trunk of a car.
According to Condé Nast Traveller, the name goes back to 18th-century horse-drawn carriages where the coachman sat on a chest, which was used to store, among other things, his boots. This storage space came to be termed as the "boot locker," which soon became simply the "boot."
How we'd say it in Britain: "I think I might have locked my keys in the boot."