If the phrase 처음 뵙겠습니다 literally means “this is the first time I’m seeing you”, why does it have 겠 in it? I thought 겠 expressed future intention. I’m asking because I like to understand the literal meanings of everything.
The meanings of the -겠- infix are not limited to simple "future" (and actually, it is only one of several future-based verb structures).
On this StackExchange, the question has arisen with 알겠습니다 already.
The meaning of -겠- in the phrase 처음 뵙겠습니다 is likely to be categorised into the third meaning there, that of "possibility / capability".
TTMIK has a lesson on it, where these various meanings are explored.
These meanings are all related. Hence linguists have dubbed it: "definite future", "deductive-reasoning", "assertive", "conjectural", "presumptive", "intentional", "prospective". Those descriptions may help, but what you really want is greater exposure to where -겠- is used in real life.