Worcestershire is the county of Worcester, a city in the west of England. Its Old English name ended in ceaster, a word for a Roman fort or town that comes from the Latin castra, a camp. The same ending is in Leicester, Gloucester and Bicester.
Over the centuries, everyday speech wore these names down. Wor-ces-ter became WOOS-ter, just as Leicester became LES-ter and Gloucester became GLOS-ter. The spelling stayed as it was.
Shire is an old word for a county. On its own it rhymes with fire, but at the end of a county name it is weak: shuh in British English, sher in American English. Worcestershire sauce, first sold in Worcester by the chemists Lea and Perrins in the 1830s, carried the name around the world.