Let me sidestep the question of how exactly entropy and the human brain are related, and instead say something about the connection between entropy and knowledge, which I think ultimately gets at your question.
An intuition you should have about Shannon entropy is that "channels" (since this is the term information theorists use) which are higher entropy require more knowledge to remember, in principle. For example, if you know that the top half of an image is white and the bottom half is black, you can write down a short formula that can reproduce the picture for you - much shorter, in fact, than the naive way to memorize an image, which is to just write down a giant table that lists the hue of every pixel individually. Consequently the "minimal amount of knowledge" required to know every pixel value is at least as short as the short description we just gave. (Another way to say this is that the image can be compressed in a very efficient way.)
However, for a specific image which was produced by a random noise process, we expect that there will typically not be a short description of the image - we can try all sorts of different "compression schemes" (algorithms which systematically look for shorter descriptions) and they will all be around the same length as just a giant matrix of pixel values. But memorizing an entire image pixel-by-pixel requires storing tens of thousands of pieces of information somewhere! This is way worse than storing a short formula or description or something. Consequently, the knowledge associated with actually knowing a specific random noise image is expected to be quite high.
In sum, entropy <-> minimum amount of knowledge required to memorize a channel (normalized against the length of the channel).
This does not directly address your intuition that indeed, a random noise image does not look like any specific piece of information to most of us. But this is the story almost every information theorist has in their head about Shannon entropy, and so is at least a decent attempt at an answer to your question.