There's no exact translation for this "잘 부탁드립니다" in English.
If you look into the meaning word by word, 잘= well. 부탁=favor. 드립니다=give(honorofic).
It might be a little bit confusing to understand.
So I will let you know when to use it with some example sentences.
You can use it in 3 different situations.
1. when you introduce yourself to a group of new people and you want to impress them.
A: 안녕하세요. 저는 서울에 사는 유진이라고 합니다. 잘 부탁드립니다.
Hi guys. I am Yoojin who lives in Seoul. I am glad to meet you all.
In this situation, you can use it as a meaning of "I am glad to meet you".
2. On your first day of work, when you want to gain a favor with your coworker (especially with a superior).
A: 사장님, 안녕하세요. 오늘부터 일하기로 한 유진입니다. 잘 부탁드립니다.
Hello, president. I am Yoojin who is supposed to work with you from today. I am looking forward to working with you.
B: 네, 저도 잘 부탁해요.
Hi, I am looking forward to working with you too.
Here, besides "Nice to see you", the meaning "I am looking forward to working with you" is included as well.
It sounds a little bit weird to say this in order to gain a favor, but Korean people will think you are nice and humble when you say this.
it's quite common to see people use this on their first day of work.
3. When you ask a favor to someone.
A: 선생님, 저희 아이 잘 부탁드립니다.
Teacher, Please take good care of my kid. (=Thank you for taking care of my kid)
B: 네, 어머니. 걱정마세요.
All right, ma'm. Don't worry.
In this situation, It has nothing to do with the greeting.
It is used when you ask someone a favor as a meaning of "I really appreciate what you are going through for me".
Do you get it? :)