The main purpose initially is for user that shifted from film(35mm) to understand the change in focal length or to make people that start with a entry DSLR that stumble upon the term "crop factor" miserable :-(
Took me quite a while to understand how the sensor size will affect the photos. IMHO I concluded that it cause more confusion than helping people understand them since most hobbyist NEVER USED a 35mm before or had more than 1 camera that have different sensor size...
To summarize, I just put up some points that I think will affect our daily usage
1. The sensor size is different, so the photo capture by the lens of the same focal length will be different (refer to image below). If you used a 80mm lens for a Full frame camera, the photo taken will be identical to a smaller size sensor photo if the lens used is divided by the crop factor. eg. A photo taken with a Canon full frame camera using a 80mm lens will have a same result as a photo taken wtih a Canon smaller size sensor using a 50mm(80mm/1.6)
Example of image capture by different sensor with same focal lenght lens
