Nothing’s moving on the screen. You’re reconstructing movement from a series of still images.

Motion blur—when it’s implemented with understanding of why it works—provides additional information about speed and direction of movement in a way that your brain is naturally apt to deal with.

Without motion blur, you need at least two frames to figure out direction of movement. With it, you’ll have enough information to start priming response in one frame. And consistent pattern of priming actions and successfully executing them translates into a physically satisfying experience.

I also dislike motion blur and turn it off because it gives me a headache, but let's not pretend that motion blur doesn't do anything at all, because it does.

Motion blur happens in real life when an object moves quickly through our field of view and the image smears on our retina. But the effect is different depending on speed and distance; a car whizzing right past you appears blurred, but watching that same car at the same speed when you are standing 200 meters away is not blurred, because the car occupies less of your field of view and so moves relatively slower across it - slow enough to not blur.

To get one thing out of the way, old LCD monitors especially (but even some newer monitors) can intrinsically suffer from what looks like motion blur, which is when the monitor can't update quick enough and moving objects leave a kind of ghosting or smearing effect behind them. High quality screens however will have very little noticeable intrinsic blurring.

Assuming a decent screen then, motion doesn't intrinsically blur - not always. Sufficiently quick events will indeed still look blurred due to natural human eyeball motion blur, but lots of things that would blur in real life might not. The car whizzing right past you won't necessarily appear blurred at all, because what in the game represents to your player character a very close and fast movement is on screen perhaps relatively much slower and further away from the perspective of the human sat at the desk looking at it on a monitor, and so doesn't read as blurred to our eyes in the same way.

So, motion blur in games is an attempt to take movements that would look blurry in real life, and apply artificial blur to make them blurry on the screen.

I don't personally like in-game motion blur, and even if it didn't give me a headache I'd still turn it off for stylistic preference reasons, but it is a real thing, and does try to achieve something specific.

Maybe if you're trying to capture still images from a video stream?