The process is roughly like this:
* GP prescribes therapy.
* You call seven public therapists who are busy and note their information and when you called. There's a specific form for this.
* You find a private therapist that satisfies certain conditions (e.g. degree granted in Germany) that is a good fit for you, and your public insurance is obligated to pay for your care.
Here's a long guide on it:
https://www.bptk.de/wp-content/uploads/2019/09/2019-09_bptk_...