On the web you will find a lot of mathematical descriptions of histogram equalizations. That's great if you want to implement it or just know how it's done. But you might be wondering how it really works, or, rather, why it works.
The key is to not thinking about percentages when you see the CDF. Think about the actual count.
Let's say you wanted to find the bottom fifty percent of a list of numbers. You could just sort them, and start collecting elements until you had half of the numbers, right? In a way, this is really what a CDF depicts: For each possible percentage from 0 to 100 %, you can read of the last element you would see if you were to do that method.
In my opinion, thinking about the CDF in that way makes it easier to see why CDF sorting and histogram equalization works.