To convince yourself that it is an illusion is easy. When the moon is full, on the horizon and looks large, put your thumb out at arms length and you will note that your thumb covers the moon. Do it again when the moon is high.
You will note that thumb covers the moon the same no matter how big or small it looks. This means that it has to be an illusion because an experiment (your thumb) confirmed that the moon is the small size in both cases. Why that is so is another question which I don't know.
It doesn't give a difinitive answer, but suggests it is partly due to a number of different illusions. The primary one, however, seems to be based on our inner ear's and eye's ability to distinguish vertical angles. Most people don't realize it because they don't know how high objects actually are, but people tend to overestimate the altitude of low objects in the sky. Because of this, we percieve the dome of the sky to be stretched at low altitude (the sky dome illusion).