To start off, HDR is an acronym for "High dynamic range".
Now, the benefits of HDR: Normally, lighting is computed with a minimum value of 0.0, and a maximum of 1.0, which is multiplied by whatever the colour of the surface or texture the light is affecting (and don't forget that when you multiply some variable x by 1, you get x). This only allows for, in comparison to the real world, a very small range of values. HDR rendering expands that range GREATLY, enabling values from 0.0 to some high number (I don't know exactly, but it's big enough). So you can have surfaces that, with a strong nearby light, become over-lit.
HDR rendering by itself is not much though, but there are some post-processing shader effects that allow HDR to really strut its stuff. The big ones that people like are light blooms, glares, and changing exposures. A light bloom is often seen as light bleeding into other parts of the image, and glares (e.g. lens flares or lens anomalies) help accentuate them. Changing exposures (e.g. pupils dilating under bright light) also enhance the realism. For examples of those three, I suggest you look at these screenshots:
High exposure (note the light bloom)
medium exposure
Low exposure
Example of a glare
What I imagine will be shown in the coming level is the effects I've outlined above, using HDR rendering.
FAQ:
Q: I thought blooms and stuff WERE HDR rendering though?
A: NO!! Like I said above, they are just shader effects. In fact, those shaders can be done without HDR rendering, and still get decent results. Are they as good as they COULD be? No, so HDR enhances the image quality of the effects by a lot. I CANNOT STRESS ENOUGH THAT THOSE EFFECTS ARE NOT PART OF, NOR ARE THEY INTEGRAL TO HDR RENDERING, THEY ARE JUST ENHANCED BY IT!!!
Q: Isn't HDR in the game already? I see glares in HL2 every now and then.
A: The "glares" already in HL2 are cheap simulations, such as placing a particle in a certain area that fades out when the player gets near them. There are also a couple situations late in the game (esp. City 17) where windows looking out onto the street are overbrightened until you get close to them. That's also part of the simulation. Both examples would have to be hand done, and are not automatic.
Q: Are there any examples of HDR rendering and usage of post process effects that I can see for myself?
A: I feel that this is the number one thing you can check out:
http://www.daionet.gr.jp/~masa/rthdribl/ It's absolutely beautiful, and despite the simplistic shapes, it all looks photorealistic (especially if you have a high-end card like an X800 or 6800, where you can have very high settings, and still retain a very smooth framerate)
Q: What other practical applications of HDR rendering are there?
A: I can't name many off of the top of my head, but I know that one application is shadow maps, since the pixel shader code for those require the high precision that a HDR texture can provide.
Q: How big of a performance hit will it be? My video card is brought to its knees by that RTHDRIBL program!!
A: It will be big, but that program is actually far more intense than is necessary at this point in time (especially with the motion blur). A good example of the performance "fall" was when Far Cry's v1.3 patch was released. 6800U's that were running the game at 1600x1200, 4xAA/8xAF and almost 60fps were shot down to 1024x768, No AA and 30fps. Part of that is due to the extra processing required for the high precision numbers, part of that is due to the post-processing effects (each of which run fairly complex pixel shaders across the ENTIRE screen).
ma kom der is tush nog veel werk aan 
