I leave.
You leave.
He, she or it leaves.
We leave.
You all leave.
They leave.
There is only one plane, so it is singular. Is not "I" or "you," so it is third person, "he, she, or it."
The plane leaves.
In contrast:
The planes leave.
(It's hard to come up with a realistic example of a situation where you would say "the planes leave." The best I can do is "At that airport, the weather the weather is often bad. Air traffic gets delayed, and all the planes leave late.")
(By the way, I sidestepped the question of whether a plane is an "it" or a "she." That's a completely different topic! Traditionally, a ship is "she," and up until the 1960s or so, planes were sometimes referred to as "she.")