Why? The APEX team implemented the login page with the HTML attribute autocomplete=off:
For a development environment I find this behaviour kind of annoying, especially when the APEX builder session times out frequently and you have to login to your workspace multiple times during the day.
Just right-click on this link and save it as a bookmark: Force browser to save the password
I have found it here, but it is explained on many other websites as well.
Once you click on your bookmark you will see the following message:
If you continue to log into APEX you will get the typical message where you can save the password now:
It is good practice to use a password for your password manager in Firefox, too. Here are the steps to enable it.
Cheers and enjoy,