To keep with the TV analogy, let's say you are watching a movie, enjoying the story, the setting, etc... Then, noised is added to the signal and you start seeing all sorts of glitches.
For example, let's imagine that the boot of a character flickers a bit. That boot has its importance, the director wanted to give the movie a certain aesthetic and that boot is part of it, but it is not important in the story and you are not supposed to notice it, in fact it is especially designed not to be out of place.
But because of the glitch, you now have all your attention focused on the boot, which is interesting, but you lost the big picture, too hard to understand now because of the noise. Maybe the next glitch will highlight the pavement or a lamp post.
The closest I can come to describing this is a higher resolution - like viewing the same photo, but you go from 640x480 to 4K, and from 256 to 32-bit color. Indeed, when consuming any sort of digital visual input, resolution differences (and compression artifacts etc) become much more noticeable, so perhaps this isn't merely an analogy. But the same also extends to sounds and even taste - it's not that things taste different, for example, but you can more readily identify and appreciate all the overtones that combine to create a particular taste, without subtracting from the experience of the whole.
This, by the way, is why watching fractal animations [1] can be so fascinating in that state - you still perceive the fractal as a whole, but you also simultaneously see the tiny details, and can observe that they faithfully correspond to the larger image, without having to individually focus on each.
[1] E.g. https://www.youtube.com/watch?v=8cgp2WNNKmQ (best viewed at 4K)
The opposite of tiny detail happens. I looked at a brick wall and all the bricks were identical. Like lazy copy paste game texture identical. I knew that the bricks weren't actually like that. You see patterns in things because the frequency resolution needed to encode the difference is cut off. The result is like a JPEG artifact but instead of localized square pixelation you get globalized crystal pixelization. Everything looks like it fits a crystal pattern. Especially random noise. Here is some noise that worked particularly well one time. https://ibb.co/D1rd6bN
You can't tell me you are seeing patters that are really there in literal randomness.
You know how trees sometimes look like they have faces, for example? They objectively do, according to our pattern recognition algorithm for faces, in a sense that people will find the same tree more or less anthropomorphic. But it doesn't mean that they actually have faces, of course! If you have a still photo of the forest, and stare at it long enough, you might notice more such than you would if you just glanced at it briefly. Same thing here.
TV static is pure randomness. Looking at any pixel has no correlation with looking at any other pixel. Any pattern you see in static must therefore be an artifact of the way your brain encodes patterns rather than something that is actually there.
For a more abstract example, if you generate a random black-and-white static image, it will have a bunch of (short) vertical and horizontal lines in it, simply because random pixel values will sometimes cluster into sequences that produce lines. Such a line is "actually there" - different people will see it, and you can even write code that finds all such lines. The fact that its pixels were formed randomly, without correlation between each other, doesn't change this - it just means that the line has no meaning; it's data, not information. But even so, your ability to detect such lines in a static image can vary.
In a similar vein, there's a well-known issue with RNG in games, where a straightforward approach can sometimes result in e.g. repeating the same value in a row many times, which the players immediately notice (and perceive as unfair). Which is why in many games, the RNG is specifically tweaked to keep track of recent outputs and actively avoid producing more output that would form an easily recognizable pattern.