No–it’s about brighter brights. You’re probably thinking that’s the same thing, so let me explain.
An HDR display requires two things: higher bit depth (at least 10 bits per component, ideally 12), and a higher peak brightness (3x at least, ideally 10x). You might be thinking–my display is already bright enough. It doesn’t need to be 10x as bright. But that’s only true of the average brightness, not the peak.
Most sets are designed to output a comfortable white level. Have a scene in a white room, and it looks pretty normal–neither blinding nor too dim.
The problem is that any colors are necessarily going to be dimmer than the room, because white is the combination of RGB, and so if you just have one component, then you’re losing the brightness of the others. Blue is the worst in this regard; perceptually, it’s only like 6% of the total brightness contribution of white.
Sometimes this isn’t a problem. A blue painted object is necessarily going to be dimmer than white surroundings. But it is a problem for anything that emits light, like a neon sign.
The answer to this is to allow color components that are brighter than their contribution to white. You make full white be (say) 10% of the possible brightness, but allow saturated colors to go higher. Or even white itself, as long as you limit it to small portions that would be dazzling in real life, like a flashlight.
Of course, if you’ve made white be at the 10% level, then you’ll have a very dim screen unless you put in a very bright backlight. So that’s what they do. That would be wasteful since most of the scene won’t need it, so ideally the backlight is segmented such that most parts can remain at the normal levels.
You also need a higher bit depth, since you’re using a smaller portion of your range most of the time. If you stuck with 8 bits, then compressing to 10% means only 25 levels of brightness, which is not nearly enough. So you add a few bits to compensate.
Back in low-dynamic-range land, designers have a choice for what to do about bright, colored objects: they can reduce the brightness (reducing their impact), or they can desaturate. If you take a solid blue object and add some partial red/green, it will get brighter and still be bluish. But it won’t be the vibrant blue you wanted; it’s more washed out. So you can have either dim saturated colors or bright desaturated colors. Both look pretty crappy. HDR allows both at once.
As an aside, you might be thinking that with all this light available, it could actually blind people if you turned it on all at once. And that’s true–so the various HDR standards have limits on average brightness. You wouldn’t want some commercial turning on the full brightness to catch your attention. It could damage both the set and your eyes.