The meaning of 잘 부탁해 is close to "Let's get along well".
to have good manners, Be kind to me ..something like that.
There's no special meaning it's like a say hello.
선생님도 잘 부탁해.
선생님 : teacher
도 : also
선생님도 잘 부탁해. (the speaker is a teacher: Let's get along well !! :)