반갑다 literally means 'to be glad' as you said, and when combined with 만나서 (만나다 = meet, ~서 = therefore, so, and) becomes 만나서 반가워요/반갑습니다 etc, which literally means 'I meet you so/and I am glad'. But colloquially, many people just shorten this phrase simply to '반가워요/반갑습니다' - we can guess the meaning if it is the first time to meet somebody. So, in short, the literal meaning is 'to be glad', but it is commonly used alone to mean 'nice to meet you', too :)

I'm sorry. I can not english very well. so I'll most easy to write a korean expression.

0. 대화에서

   - 반갑다,반가워 : 본인보다 어린 사람 혹은 친구에게 하는 인사말

   - 반갑습니다, 반가워요 : 서로에 대해 잘 모를 때 예의를 차려서 존칭해주는 인사말 

0. 결론적으로 의미는 같지만 쓰임은 다르게 사용해야합니다. (It means both but you have to use just one)

Soymylk good job!