Can anyone explain why is it grammatically correct to say "off you go"? Well, it seems really strange to me.
Some adverbs, including directional ones, can be put first in the clause, for emphasis or rhythm:
Down they fell, all the way to the bottom.
Away he sped, pedals whizzing round.
Off he ran.
With a first or second person subject it is usually an exclamation, but grammatically it is the same as with any other subject. Verbs other than "go" don't seem to fit very well in this case:
In I go!
Off you go!
Away we go!