Why 'doing' after 'look forward to'?

The key to understanding this usage is the preposition “to” which comes after the expression “look forward:”

Look forward to something means to be pleased or excited that it is going to happen. The ‘to’ in look forward to is a preposition, so we must follow it by a noun phrase or a verb in the -ing form:

• I’m looking forward to the holidays.

• A: Are you excited about your trip to South America?

• B: Yes, I’m looking forward to it.

• We’re looking forward to going to Switzerland next month.

If the second verb has a different subject, we use the object form of the pronoun, not the subject form:

• We’re looking forward to him arriving next week.

Not: We’re looking forward to he arriving next week.

(Cambridge Dictionary)

"look forward to" takes a noun. "drink" is a verb. So we take gerund "drinking", which acts as a noun.