Given that light spreads out as it travels, why do distant objects appear smaller than nearer objects?

Spreading of the light emitted by a small part of an object implies that less light from that region can enter the pupil of an eye (or of an optical system). Therefore, it affects the light intensity.

Instead, the apparent size of an object has to do with the angle, at the position of the eye or other detector, made by rays coming from different emitting regions of a distant object.

For this to understand lets take an example of the apparent position of stars . Light reaches to your eyes after bending all along the path in the earth's atmosphere.

And we see the star along the straight line of the final light which comes to your eye . And this is how we see everything around us.

This picture might explain my texts in short

Now , suppose there is a point source $P$ . Since the light rays spread from that point and it's shape is in the form of a cone (you can visualise).

Suppose all the light rays enter your eye. Since you see objects along the same line of each ray which enter your eyes (as explained above) , go back along each rays of the cone , What do you get ? Do you get the one represented below as yellow dotted circle in the right ? Or point $P$ ?

I think you got your answer.

Hope it helps ☺️.