I agree that the English translation is not grammatically correct; a better one would be "It's not the animals who get into the airplane, but the people."
Should be: The animals don't get on the airplane, the people do.
The English translation of this is rather awkward. We would more say... "The animals do not get into the airplane, but the people".
"Not the animals" sounds odd in this context.
I think they wanted to stress that THE ANIMALS, as this is what the Hungarian sentence means - the animals in focus. So, it is a literally translated sentence that makes more sense in Hungarian than in English.