Since none of the answers above actually explain what happened, I decided to chime in and bring some more details to this issue.
Yes, the solution is to run the MySQL Upgrade command, as follows: mysql_upgrade -u root -p --force, but what happened?
The root cause for this issue is the corruption of performance_schema, which can be caused by:
This issue might have been present on your database even before the patch, but what happened on MySQL 5.7.8 specifically is that the flag show_compatibility_56 changed its default value from being turned ON by default, to OFF. This flag controls how the engine behaves on queries for setting and reading variables (session and global) on various MySQL Versions.
Because MySQL 5.7+ started to read and store these variables on performance_schema instead of on information_schema, this flag was introduced as ON for the first releases to reduce the blast radius of this change and to let users know about the change and get used to it.
OK, but why does the connection fail? Because depending on the driver you are using (and its configuration), it may end up running commands for every new connection initiated to the database (like show variables, for instance). Because one of these commands can try to access a corrupted performance_schema, the whole connection aborts before being fully initiated.
So, in summary, you may (it's impossible to tell now) have had performance_schema either missing or corrupted before patching. The patch to 5.7.8 then forced the engine to read your variables out of performance_schema (instead of information_schema, where it was reading it from because of the flag being turned ON). Since performance_schema was corrupted, the connections are failing.
Running MySQL upgrade is the best approach, despite the downtime. Turning the flag on is one option, but it comes with its own set of implications as it was pointed out on this thread already.
Both should work, but weight the consequences and know your choices :)