[bizerl (OP)]

I'm almost embarrased to ask - it seems like such a simple question and a concept that we take for granted.

But what exactly is the mechanism that reduces the intensity of an image the further it is away from us?

[RD]

For "why less bright" see ... https://en.wikipedia.org/wiki/Inverse_square_law

[CliffordK]

The apparent size of objects is essentially the same. 

You can consider the viewing angle.
If you calculate the angle from your pupil to each edge of the object in view. 
The closer the object is to your eyeball, the wider the viewing angle.
The further away, the narrower the view angle.
And, thus the smaller the object appears.

Or, you can think of your visual field as describing an fixed viewing angle (an isosceles triangle).   The further away, the larger the "base" of the triangle, and thus the more stuff that can be fit into the visual field, with each item appearing smaller.

[Lab Rat]

Couldn't this be somewhat compared to the Doppler Effect?

[wolfekeeper]

For "why less bright" see ... https://en.wikipedia.org/wiki/Inverse_square_law

That's not right. The size goes as the inverse square as well.

So it turns out that the intensity of an object is independent of distance, provided that there's nothing in the way that scatters the light, for example dust or clouds.

That why you can see stars, even though they are immensely far away.

But obviously things get smaller with distance, just not any less intense.