This is a New Year's phrase not-so-commonly heard in Deitsch but very commonly heard with relevant cognates in German. It's meaning literally is, "Good slide!"
Wishing one a good "Rutsch" has multiple meanings in Deitsch, though... Some are quite insulting (someone, usually in a sexist structure but may be directed at anyone, with loose morals), some area a little funny (a squirmy kid), and then there is the intended meaning (slide).
Thus: Gudi Rutsch ins Nei Yaahr... A good slide into the New Year!
(This would have a very different meaning from "Gudi Rutsch im Neie Yaahr," which would be wishing one a good physical sliding board, squirmy child, or person with loose morals in the New Year).
It is also worth noting that "Rutsch" is grammatically feminine (die) in Deitsch but grammatically masculine (der) in German.
Grammar matters! :)