This means you spit on something and use the spit to shine it.

Shoe polish is used to shine shoes. Normally, the last step is to take a cloth and "buff" (rub the shoe aggressively with the cloth using a "snapping" motion) the shoe so that polish shines. Many years ago, it was realized that if you put a small amount of certain liquids on the shoe during this stage, the shine was intensified so the shoe would cause a reflection like a mirror. The easiest and most available liquid to use, and the cheapest, was spit. Hence, spit-shine. Today, other liquids are used since spitting on a shoe seems a little crude to some. So, anything that is "spit-shined" is polished, neat and shiny more so than normal. It does not mean that the uniform actually went through the process of spit-shining, just that the soilder took extra care to make sure it looked the best that it could.