뵐게요 means (I)'ll see (you), but I am struggling to see how to arrive in this form.
Applying the most general verb conjugation rule for 해요체 to 뵙다 (to humbly meet/see), I expect to get 뵙어요. But if we add -겠다 to express future tense, then I expect to get 뵙겠어요. So how does it arrive at 뵐게요?
뵙다 is an unusual verb. It's pretty much the same as 뵈다 (except maybe a little more polite), but unlike 뵈다, 뵙다 can only combine with a suffix starting with a consonant other than ㄹ. (But see below for clarification.) From Naver dictionary:
뵙다
(자음 어미와 결합하여) (사람이 웃어른을) 대하여 보다. ‘뵈다1’를 더욱 공손하게 이르는 말이다.
So, if 뵙다 were a completely regular verb, then its conjugation would be something like "뵙을게요", but it's actually a wrong form - nobody speaks like that. The form actually used, 뵐게요, is a conjugation of 뵈다 (i.e., 뵈- + -ㄹ게요). (An example of a regular verb is 잡다 -> 잡을게요, or 입다 -> 입을게요.)
I think it's because 뵙다 is actually short of 뵈옵다, which is just 뵈다 plus an old-fashioned -옵- suffix (extra politeness). Like 뵙다, -옵- is restricted by the following sound - when it's not allowed, an alternate form -오- is used. (You can still find the use of -오- in some stock phrases like "... 출입을 제한하오니 양해를 부탁드립니다.")
Now, actually, the rule is a bit more complicated. (You probably noticed that "하오니" ends with "니", which starts with ㄴ.) I think you can basically "test" it by comparing with other irregular verbs like "걷다 (walk)". If a suffix combines with "걷-", then it can combine with both 뵙- and 뵈- (or -옵-): 걷고/뵙고/뵈고, 걷자/뵙자/뵈자, 걷기/뵙기/뵈기, 걷던/뵙던/뵈던. On the other hand, if a suffix combines with "걸-", then it can only combine with 뵈- (or -오-): 걸으니/뵈니, 걸을게요/뵐게요, 걸었다/뵈었다, 걸어요/뵈어요, 걸음/뵘.
