Why isn't Yahoo keeping me signed in even though I've checked the option on the box?

If you can't stay signed in your browser isn't storing cookies from Yahoo. You have to change your browser settings to allow them.

Your browser may be set to clear all saved passwords when you clear your cache and cookies, your passwords may be erased when you clear your browser History or your browser may be automatically clearing your cache and cookes when you log out or shut down. Follow the steps to adjust your browser settings:

Browser clears Yahoo settings automatically

https://help.yahoo.com/kb/sln4651.html

You may need to enable your browser's Autofill feature to remember your Yahoo ID and password. Follow the steps to enable Autofill:

Use Autofill to save your Yahoo sign-in information

https://help.yahoo.com/kb/sln2164.html

If someone else uses the same computer or device as you, they may be logging out when they use Yahoo.

Your PC I guess