"Nice to meet you. I am Minji."
Translation:처음 뵙겠습니다. 저는 민지입니다.
처음(으로) - for the first time, 뵙다 - most formal version of 보다. However, 뵙다 in conjugation can only use 겠다 (and maybe something else, someone correct me if so). So the first sentence is saying "We're seeing each other for the first time." I've never heard someone say that before though. People usually say "반가워요" (I'm glad [to meet you]) or "만나서 반갑습니다" (Since we're meeting, I'm glad).
I agree, I haven't heard 처믐 뵙겠습니다 very often either (though I have to admit I don't have a lot of contact with Koreans who would warrant this level of respect xD). I wonder if it has a long native history or if it's a calque on Japanese 初めまして.