HDR stands for High Dynamic Range, meaning that an HDR enabled monitor or TV can support a much wider output of colour and brightness levels than a normal standard dynamic range TV can. HDR enabled games just support this standard, so they output a wider colour range. Essentially dark parts will look darker, bright parts brighter and colours in general should have a bit more depth in them; though all of this depends on the quality of the monitor or TV, as not all TVs marketed as supporting HDR truly support the 10-bit colour output and improved contrast levels required to do it justice.
You'll still benefit from the increased resolution of the games with Pro support patches however, as well as framerate improvements in general and a few graphical boosts here and there. With or without HDR, the PS4 Pro will give you a fairly decent graphical upgrade over the base PS4 if you have a 4K TV or monitor.