- R.S., Saco
You are referring to the well-known "moon illusion," when the moon certainly does appear to be at its largest. Actually, the moon is no larger when it is on the horizon than it is when higher in the sky. You can verify this assertion for yourself. When you see the moon on the horizon, you'll find that you can cover it with your thumb extended out at arm's length, just as you can when the moon is higher in the sky. The moon's average angular diameter is about one half a degree, the same as the Sun's.*
You are correct that the moon's distance from Earth does change throughout its orbit. It varies from a maximum (apogee) to a minimum (perigee). The perigee moon is about 13% larger than the apogee moon. However, the perigee of a given orbit is separated by the corresponding apogee by about 13-14 days. (The period between successive perigees - called an anomalistic month - is 27.5 days.) Realize, also, that when we're seeing the moon on the horizon, others in different locations are seeing it in a different position in the sky.
Many astronomers have attempted to explain the moon illusion. One popular explanation stated that the moon appears larger on the horizon to us because we're seeing it in relation to large objects with which we're already familiar, such as trees, houses, and cars. However, the enlarged horizon moon effect has been seen by desert dwellers and jet pilots, as well.
The currently accepted theory involves the "Ponzo illusion."
This illusion, named for the Italian psychologist Mario Ponzo (1882-1960), is caused by the way the mind perceives objects relative to their background. The moon appears large on the horizon because something in our minds causes such objects to appear enlarged.
We should remember that the moon is not as close or as big as many people like to think. Earth's diameter is about 7900 miles and the moon's is about 2150 miles. Its average distance is about 240,000 miles. On a scale model, if Earth were a basketball, the moon would be the size of a tennis ball about 24 feet away.** That distance is about one quarter of a regulation basketball court. Even though the moon is the closest celestial object to Earth, it's separated from us by quite a lot of space.
As usual, I hope this answer proved helpful and didn't make matters worse.
*Greetings! This bothersome aside has been added to inform you that the moon's diameter will not always equal that of the Sun. The moon is receding from Earth at about 3.8 cm per year. Eventually, the moon's apparent angular diameter will be so small that total solar eclipses will no longer occur: however, we can expect total solar eclipses to continue for hundreds of millions of years.
**We're using the English system of units in honor of today's UK general election and not because I am too pathetically lazy to pencil out the conversions.