Checking the "Remember my username only" box saves your Username on your computer. That means you don't have to enter your Username every time you sign in. You only have to enter your Password.
Please note: Do not save your Username on a public PC. When you save your username, it is stored on the computer you are using at the time. If you do save your username on a public PC and want to remove it, uncheck the "Remember my username" box, sign in again, then sign off.