Because each stop must be double the value of the previous stop. Two raised to the power of eight (2^8) is 256. 0-255 is 256 distinct values. Half of 256 is 128. In an eight-bit system the value 127 is one stop dimmer than 255. The value 63 is two stops dimmer than 255, and so on. You need to be able to double the dimmest value to have a one-stop range, because if everything were the same value then you have no stops of difference between the brightest and dimmest parts.

For purely mathematical reasons, 8-bit jpegs aren't capable of ten stops of dynamic range. They're theoretically capable of seven stops. In the real world it's between six and seven. That's one reason why raw files must be processed and light curves applied to squeeze the 10-12-14 stops of dynamic range possible in a 12 or 14 bit raw file into the 6-7 stop capability of an 8-bit display device.

A scene that is four stops brighter than another scene is not four times brighter than the other scene, it is sixteen time brighter (2^4 = 16). A scene that is 10 stops brighter than another scene is not ten times brighter, it is 2^10 = 1,024 times brighter. This means that for every photon per unit area in the dimmer scene, there are 1,024 photons striking the same unit area in the brighter scene.

So to clarify, if I had a scene with really light parts and really dark parts, a camera with good DR would be able to capture the detail in both the light and dark parts. Where a camera with bad DR would either get rid of the shadows or highlights depending on the exposure because the difference in light from the brightest and darkest object was too much.

Pretty much every interchangeable lens camera on the market today can capture a higher dynamic range than jpeg can display. It's as much about understanding how to get from the raw sensor data to a finished JPEG image, whether one is processing raw files using the camera's settings, many of which are adjustable by the user before taking the shot, or using a post-processing application on a computer to process the raw files, as it is about the exact dynamic range a particular camera/sensor is capable of capturing.

For more about the difference between raw files and JPEG images, please see:
RAW files store 3 colors per pixel, or only one?
While shooting in RAW, do you have to post-process it to make the picture look good?
Why can software correct white balance more accurately for RAW files than it can with JPEGs? (There are a lot of additional links to other questions here at Photography SE in the accepted answer that you might find helpful)
What does an unprocessed RAW file look like?

For more about how to reproduce high dynamic range scenes using lower dynamic range display mediums, please see:
What's the difference between "Fake HDR" and real, bracketed exposure HDR?
What's the point of capturing 14 bit images and editing on 8 bit monitors?
Is tone mapping automatically applied to RAW images?