Revealing mistake: When Kong is about to attack the train, the building on the right (a superimposed projection) wobbles like jelly for unknown reasons. It's not because of Kong's footsteps, for it never happens in any previous scenes.
Continuity mistake: When Kong lifts Ann from the bed she is wearing a short dress. In the close-up the dress goes way down to her ankles.
Continuity mistake: Kong breaks the window and Ann starts to shout and extends her right leg. A shot later it's bent downwards.
Continuity mistake: Kong breaks the window and takes Ann from bed. In the wide shot (where she's replaced by a stop-motion doll) she faints backwards. In the immediate close-up she is conscious and upright.
Continuity mistake: After Kong escapes from the theater, a car crashes against a wall with no glass on the windshield. In the next shot, broken glass appears.
Revealing mistake: At the Empire State, Kong extends his arm to grab the woman. Problem is, instead of catching her she magically flies up the air to his hand.
Revealing mistake: At the Empire State, the woman is an obvious square-shaped footage of film pasted on top of the footage of Kong. After the plane attacks Kong and his right arm goes limb it goes behind the pasted footage.
Answer: It's never shown or explained, and the film uses a broad "suspension of disbelief" premise. The audience just accepts the characters were able to somehow transport a huge ape to New York City.
raywest ★