HDR stands for High Dynamic Range, which sounds geeky (and it is) and scary (which it isn't). In short, it is a technique that can create photos that match the ability of the human eye to see extremes of light and dark (i.e., the range part of high dynamic range) at the same time—an ability that cameras actually lack!
Why should real estate agents concern themselves with HDR? There's no doubt that people are visually-inclined. Assuming two listings are practically the same in price and features, the listing with the better photos is likely to get the first consideration. No one wants to see a home that looks like a dark dungeon. With the right tools (or photographer/artist), an HDR photo taken at dusk can make even your least expensive property look like a magical castle. I certainly can't afford a million-dollar home, but I'll often look at agents' listings that use HDR photos just to see how amazing the images can look. Our own office's Wyatt Poindexter uses HDR photos in many of his listings.