'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. :)