“I am looking forward to seeing you” is correct. In this case “to” is a preposition, and you always need to use the ing form(gerund) after a preposition. “I am looking forward to see you” is incorrect.
“To look forward to” is a phrasal verb
To look forward to something is a multi-word phrasal verb.
A phrasal verb contains a verb and a preposition and in this case, two prepositions(forward and to)
I am looking forward to spending the summer by the beach.
To look forward to is a transitive verb which means that it needs an object( a noun or a verb).
Other examples of transitive verbs are like and enjoy. You can’t say “I like” without an object.
I like tennis.
I enjoy swimming in the sea.
Gerund after a preposition
There is a rule in English that says that if you use a verb after a preposition, then the verb needs to be in the ing form(gerund).
Thank you for buying me breakfast.
He always dreamed of going west.
After seeing the doctor, I went home.
“look forward to” is a phrasal verb, so to is a preposition in this case. Try to remember that “look forward to” is always together and a set phrase.
More phrasal verb examples(Notice the ing form after the phrasal verb)
He started out cleaning the toilets.
John ended up renting an apartment next to me.
That criminal will pay for being so bad.
Different uses of “to”
The confusion between the difference between “I’m looking forward to see you” and “I am looking forward to seeing you” is because of the word “to”
After many verbs, it is necessary to use the infinitive(to + verb) or the gerund form(ing) of the verb for the second verb.
As explained above “to look forward to” is always together. It can’t be separated.
Examples of infinitives
I want to go to the supermarket
I tried to open the bottle.
Examples of gerunds
He likes swimming in the sea.
He enjoys dancing by himself.
“To look forward to” is an example of a verb that takes a gerund.
I am looking forward to dancing all night long.
I am looking forward to taking off my shoes.
I am looking forward to playing my guitar later.
I am looking forward to meeting your family.