If you are seeing a image like this, either your network connection is slow, or you have a Google Play games log in issure. Sometimes, all you have to do is wait. If you don't want to wait, you can click the button cancel, and play the game as Player ID 0. This won't hurt anything.

How to fix Google Play games cannot log in issue

Step 1: Make sure your internet connection is strong

Make sure you can access Google Play, either through the app or the website.

Step 2: Make sure your have the latest Google play Apps

Make sure you have the latest Google play services & Google play store and Google play games. Please download and install the latest Google play Apps.

Step 3: Make sure Google Play Games app is enable

If you see that the Google Play Games app is disabled, make sure to press the Enable button to activate it.

Step 4: Clear app cache and app data

Step 6: Reboot and reinstall

Check to see if you are now about to sign into your Google Play Games account, and head over to the next step if you are still facing issues.

Step 6: Complete your Google+ profile

If issues persist, you can also try completing your Google+ profile.

Once the process is completed, you will most likely be able to access the Google Play Games service in other apps as well.