What does it mean for an image to have a distribution?

First, we must understand that an image can be represented as the numerical values of its pixels. For example, a 256x256 grayscale image can be represented by a vector of 65536 integer values between 0 and 255.

Now, let's consider that we are talking about images of human faces, like the CelebA dataset, but in grayscale.

And finally, let's imagine that we sample a vector of 65536 components from a uniform distribution between 0 and 255, that is, from $x sim U[0, 255]^{65536}$. If we interpret such a randomly sampled vector as an image, like the interpretation we described in the first paragraph, do you think that is it probable that we see a face? Well, probably not. Probably we would see just noise, like this:

Why do you know that we won't see a face? Well, because you know that a face is not composed of uniformly random pixels, but has some structure in it. In other words, because you intuitively know that the distribution of pixel values of a face image does not match $U[0, 255]^{65536}$, but has a different distribution.

As a summary, the distribution of an image domain is the distribution the pixel values follow. Note that this is applicable not only to grayscale images, but to color images expressed in the RGB space or any other numeric space like HSB.

P.S.: this is how an image of Gaussian noise looks like (it was generated with Imagemagick, with convert -size 256x256 xc:gray +noise Gaussian out.png):