The "weigh-hey" part has many variations, so it's more about the sound than any meaning in the lyrics (I sang it in school as "hoo-rah".)
I guess the "up she rises" part refers to hoisting sails, but there's very little information on how this song came about. It's just a jolly shanty to sing.
If you ever say or write this phrase, we'll all immediately start singing "What Shall We Do With The Drunken Sailor?" in our heads. As I am now.