만나다 = meet
반갑다 = be glad/nice
Therefore, 만나서 반갑습니다 = Nice/glad to meet you.
You can say both of them when you first meet someone, the someone will know it implies '만나서' even if you say only 반갑습니다. But if I say any of those to my friend, the friend will think I must be crazy.
You can use -서(ex. 좋은good → 좋아서, 따뜻한warm → 따뜻해서) when you express the causal relationship. It will be easier to understand if you think the meaning of '그래서so'. It's very often used in everyday Korean language, because it's lighter/shorter than - 때문에 or - 까닭에.