akemashite omedetou gozaimasu
(The last year) has ended and turned into New Year, congratulations!
Happy New Year
あけまして comes from the verb “明ける” meaning “(something) ends and turns into a new state”, e.g 夜が明ける (night turns into day); and おめでとうございます means “congratulations”.
Among young people, it’s often abbreviated as あけおめ.
It’s usually followed by another greeting 今年もよろしくお願いします (Let’s build/keep a good relationship this year, too). It can be also abbreviated as ことよろ, but it sounds more casual/slang than あけおめ.
Happy New Year! Let’s keep a good relationship this year, too!