Why do objects appear smaller when farther away?

Apparent size is not measured as an ordinary size, in meters. It is actually an angle, so it is measured in degrees or radians.

See this picture:

Apparent size as an angle

The object on the left is the eye. Looks like as the object moves further, the angle becomes smaller. That is what is called perspective.

Sometimes people try to compare apparent size (solid angle) and real size, but that makes no sense because they have different dimensions. For example, I've been asked:

Is the Moon bigger or smaller than a 1€ coin?

The answer is that it is much, much bigger: about 3000 km vs 2 cm. What the question is trying to ask is compare the apparent size of the Moon with the real size of a coin, and that makes no sense. You should compare the apparent size of the Moon with the apparent size of the coin, but then you should say what distance the coin is.

For reference, the Moon apparent size is about half a degree. That is about the size of your thumbnail, with the arm extended. It does not matter if your hand is big or small because a big hand will also mean a big arm!