There is an added dimension to this, and it is the ~100 miles of atmosphere that is between you and the moon, as well. The Moon is, on average, about 240,000 miles away. That's about ten times a trip around the Earth's equator.
When the sunlight reflects off the Moon and heads to the Earth, it passes through our atmosphere. If the weather is right, that air can have a magnifying, spreading, or diffusing effect that makes the Moon appear bigger to your brain. Your brain then compares the apparent size of the Moon at 240,000 miles away, to the sizes of trees on the horizon, maybe 10 miles away at most, and thinks it knows how big the Moon is based on things you normally can touch.
Since the Moon is 24,000 times farther away than the tree, and much much much much bigger to begin with, your brain is fooled into thinking that the Moon looks bigger than it is. And then another part of the brain calculates the apparent brightness and basically does it incorrectly. Again, this is because your brain has no real value of how bright the Moon really is, it only knows how bright things are compared to each other.
Combine this with the atmospheric distortion, and your brain basically calculates an apparent brightness of the Moon as something more than what it would be if someone in space, 240,000 miles on the other side of the Moon, were to try to calculate the same brightness when the Moon was full from their perspective.
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