Why does 三明治 mean sandwich when 三 means 3, 明 means bright/clear, and 治 means to rule?
That 三明治 came from transcribing the English word Sandwich into Chinese. That is to say, it is meant to approximate the pronunciation of the English word. You aren't meant to interpret the individual characters literally.

As Stan points out, it is sometimes transcribed as 三文治 as well. Historically it has also been written as 三味治, but that's pretty much extinct.