Who is using my worker threads? SQL Server 2014 - HADR
Community wiki answer:

You have a high number of databases in availability groups, that's going to be where your threads are going. There is a lot involved in the compression, encryption, and transport cost. Try turning off compression, it will reduce your thread usage by a about a third (depending on the replica count).

The question is tagged SQL Server 2014, which will by default use compression. SQL Server 2016, by default, will not use compression for sync.

You may need to increase the worker threads on the instance, or better: balance out the most active ones and inactive ones on multiple servers. See the related Q &amp; A AlwaysON Availability group query very slow. 

You may also find it is an application that is unable to close requests properly. This can result in lot of sleeping sessions lying around (which consume workers).

The number of threads actually used depends on how active the databases are. You could have 1,000 databases and, if most are idle 95% of the time, you won't have any issues. It seems that your databases have become active more often and have eaten more of your threads. That's the long and short of it.