Open the Solflare app and click the settings button ⚙️ on the bottom right.
Click on your account at the top of the screen, make sure the desired account is selected, and click EDIT.
Click the Settings ⚙️ symbol:
Press "Export recovery phrase"
Make sure no one is looking and click Show My Recovery Phrase.
Manually write or import the phrase into your desired location.
You've successfully exported your recovery phrase which has access to nearly an infinite amount of private keys.