Explanation of 또 뵙게 돼서 반갑습니다? Why is there 게 and 서?
Explanation of 또 뵙게 돼서 반갑습니다? Why is there 게 and 서, are they particles?

Also, the 겠 in 처음 뵙겠습니다???

I know the meaning of each sentence, I just don't understand where the mentioned syllables come from and what purpose they serve.

Question

Explanation of 또 뵙게 돼서 반갑습니다? Why is there 게 and 서?
Explanation of 또 뵙게 돼서 반갑습니다? Why is there 게 and 서, are they particles?

Also, the 겠 in 처음 뵙겠습니다???

I know the meaning of each sentence, I just don't understand where the mentioned syllables come from and what purpose they serve.

Answer

'게' and '서' are connective endings. 
The basic sentence structure is '-하게 되어서/돼서'. 
It is combination of '-하게 되다' and '-어서'. 
'-하게 되다' shows some situation what you get into or something what happened to you. '-어서' shows the reason, cause and grounds of what happened. So, in your sentence, "또 뵙게 돼서 반갑습니다" is showing you are in a situation meeting someone again and that is the reason you are glad. :)

'-겠-' is a euphemism and is used to avoid a flat expression. :)

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