From my experience, this occurs a day or two after a password change. The steps below will resolve this problem with Outlook 2019, but I'm not sure about 2013. Below is a step-by-step guide I have used with my computer and computers of my co-workers. If you have any questions, let me know.
Email Issue: After you’ve signed in and checked the “remember me” box, the email still asks for your password. This not an uncommon problem and the fix is fairly simple.
Spoiler alert: this requires a REBOOT, so you’ll probably want to close everything else out.
Step 1:
Go to the control panel (click the windows button and start typing “control” (even if there is not a text box there) and it will pop up in the menu).
Step 2:
Go to “Credentials Manager”
Step 3:
Click on the “Windows Credentials” tab
Step 4:
Under the Generic Credentials section, remove all of the credentials. Don’t worry about removing them; Windows will repopulate them on reboot.
Step 5:
Reboot the computer. (Note: If the computer wants to reboot in Safe Mode, go ahead and let it, then reboot again after the final step and it should reboot normally.)
Step 6:
After reboot, open Outlook and click on the “File” tab
Step 7:
Click on the “Account Settings” box (first one), then click “Account Settings” on the dropdown menu
Step 8:
Make sure your email is highlighted and click the “Change” button.
Step 9:
On the next screen, click the “Next” button.
Step 10:
Try signing in again. It should work this time and, if you clicked the “remember” box, it should not prompt you again.