Any web-based email service should always require you to log in. However you can configure your browser (rather than GMX) to remember your password if you so wish. (If you don't know how to do that, please tell us which browser you're using).
I can't find any way to get GMX to then go straight to your Inbox. (That's probably because they want you to see the ads which pay for the service).
The only way I know of getting automatic access to both GMX and your inbox is to use a proper email client (such as Outlook Express, Windows Live Mail, Windows 10 Mail, Thunderbird, etc) to send and receive your mail, which is really the 'proper' way to handle email. (Web-based services came later and will always have limitations when compared to using a dedicated email client).
If you want to configure an email client to use GMX you can find the instructions here:
https://help.gmx.com/en/applications/imap.html#servers