Which one is correct?
I look forward to meet you during the coming festival.
or
I look forward to meeting you during the coming festival.
Please help me with explanation.
There are some set rules, avoiding them will make is sound unnatural or incorrect.
The pattern with look forward to is
Look forward to + Noun
Look forward to + verb-ing
These are set pattern, not following them will make the sentence incorrect.
The to in this expression is the preposition, not the 'infinitive marker'. Consequently it requires an NP—a noun or a noun-like term—as its object.
In this case the object we want to use is an entire clause: “I meet you during the coming festival”. There are three ways† of transforming this into a construction which may act as an NP: an infinitive clause, a that clause or a gerund clause:
infinitive clause: ... me to meet you during the coming festival.
that clause: ... that I meet you during the coming festival.
gerund clause: ... me/my meeting you during the coming festival.
However, English verbs are picky; each verb has preferences and ‘licenses’ (permits) only specific types of clause for its complements. In the case of look forward to, only the gerund clause is licensed, so that is what you must use.
I look forward to [me/my meeting you during the coming festival].
There is a final operation before this becomes idiomatic: when the subject of the complement clause is the same as the subject of its head clause, it is deleted. That is the case here, so me (or my) is deleted:
I look forward to
me/mymeeting you during the coming festival.
I know of no reason why look forward to doesn’t like that clauses; it just doesn’t. But it is a general rule that phrasal verbs whose last piece is the preposition to don’t take infinitive complements, because that would put the two tos next to each other, which would be confusing:
∗ I look forward to to meet you during the coming festival.
† There is a fourth way, a free relative clause; but it is used to make a specific constituent of the clause the object, not the entire clause, so I ignore it here.
∗ marks an expression as unacceptable
look forward : to anticipate with pleasure or satisfaction <looking forward to your visit>
; Dictionary.com isn't much help, neither here nor here. I'm all for exhorting people to use the dictionary, but I don't think this question is "easily answerable with a dictionary."