If you are a web browser and you set your monitor to 6500 K then you will see images pretty much the way they come from the photographer. If you set the monitor to 5500K, then all the images will have a warm brownish tinge to them. But if you are a printing house or graphics art person, the standard lighting around the world is for lights of 5500K. So if you look at your monitor at 5500 K and you look at your print at 5500K the colors will be a closer match, and adjustments in the editor will be easier to make. And people who print for a living use a different computer for browsing the web.
When I print - which isn't often these days - I look at the prints under a 5500K lightbox, and set the monitor to 5500K. It only takes a short while to get used to the color as the eyes adjust quickly. But some colorimeters restrict the color temperature that you can calibrate to, so if you intend to both browse and print you need a colorimeter that will allow you to choose both 5500 and 6500 K settings so that you can switch back and forth from the monitor preferences panel.
The temp 5500K was chosen as being closest to the color of noon daylight - I don't know where in the world they measured this but I suspect the French had something to do with it. In South Carolina in the summer it is definitely a cooler light.
Of course if your viewers look at your prints under these new fangled low energy bulbs, it doesn't matter what color temp you use - it's going to look like ....