The sauce is one of the many legacies of British contact with India. While some sources trace its usage in Europe to the 17th century, it became popular in the 1800s.
Legend has it that Lord Marcus Sandys, ex-Governor of Bengal (and a figure unknown to history outside this tale) encountered it while in India in the 1830s. On his return to Britain from the subcontinent he came to miss the condiment, and in 1835 he asked John Lea and William Perrins, chemists and pharmacists from Worcester, England to make him some from a recipe he had brought with him.
Upon completing the necessary steps, the resulting product was so strong that it was considered inedible, and a barrel of the stuff was exiled to the basement. Looking to make space in the storage area a few years later, the chemists decided to try it once again (possibly to see if it was as bad as they remembered), only to discover that the sauce had mellowed and was now quite palatable. In 1838, the first bottles of "Lea & Perrins Worcestershire Sauce" were unleashed on the general public. It was a considerable success, and both the condiment and Lea & Perrins are going concerns to this day.
see also the Wikipedia Cookbook