"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!
開ける・あける is a transitive verb, 開く・あく is an intransitive verb. 開く・ひらく can be both transitive and intransitive.
あく is to open by moving something like a key or a lid that has blocked a passageway. The movement is in one direction only. This word is used for things like doors, windows, or bottles.
The window opened. (Note this is the intransitive version so the focus is not on how the window was opened, just that it opened.)
ひらく is more of a blooming effect, where something opens outward from center. This word is used for things like flowers, hands, umbrellas, or automatic doors. (If you were talking specifically about a flower blooming, you'd use the verb 咲く・さく, to bloom. 花が咲きました。The flower bloomed.)
Please open the umbrella.
Sometimes either word can be used, such as for eyes or mouths.
The eyes/mouth opened.