What does my zip code plus 4 37398-4477 mean?

A ZIP+ 4 code uses the basic five-digit code plus an additional four digits to identify a geographic segment within the five-digit delivery area, such as a city block, a group of apartments, an individual high-volume receiver of mail or any other unit that could use an extra identifier to aid in efficient mail sorting
373 = somewhere in the vicinity of Chattanooga, TN
37398 = Winchester, TN (including some outlying areas, but not including Estill Springs, Belvidere, Huntland, or Cowan)
44 = a chunk of 37398
77 = Randy's Lazy Dog Saloon

The last two digits normally identify one side (even or odd) of one block of one street, but most businesses have unique ZIP+4 codes.