"Please open the gate."
Intransitive verbs don't have an object (target of the action), and transitive verbs do.
Here you don't ask the door to open by itself, but to someone to perform the action of opening on something (the door), and that something is the object. So, as the verb (open) has a target/object (door), it is transitive (because it transitions the action into the object)
There are two verbs to say "open" in Japanese, one is あく, which refers to something opening by itself (e.g "the door opened") and the other is あける, describing someone opening something (e.g "someone opened the door"), which is what we need in this sentence. あける is あけて in the -te form, so: あけてください. Hope this helps!