Looking forward to see you vs Looking forward to seeing you?

There’s two different things going on here, both of which use the word to, which is probably what’s confusing you.

The rule your teacher taught you applies to infinitives, in the context of sentences where there are two verbs, like

I like to run.

The verb following to is in the present tense, as is expected. In this case, to is known as a “particle,” which is basically a word that doesn’t fit into nice grammatical categories, but has some meaning.

I like to running.

Is not correct.

However, what’s happening here is that to is a preposition connected to the adverb forward, which is modifying looking.

You look forward to nouns. These sentences are all acceptable:

I look forward to my wedding.

I’m looking forward to the weekend.

I was looking forward to this weekend, but I got sick.

So in this case, you have to use a gerund, the noun form of a verb following forward to. In English, the gerund is identical to the present progressive, so you get sentences like

I look forward to seeing you.

I look forward to meeting you.

I’m looking forward to dogsledding this winter.

Each of these sentences are acceptable, and use a gerund (verbal noun). You can’t use other forms of the verb after the preposition to, you can’t say:

I’m looking forward to see you.

I’m looking forward to saw you.