After a transcription is complete, click on the More Actions menu (⋮) 

The Copy text option will copy the text of the current transcription to the computer's clipboard. From there you can paste it into your own document or your genealogy software.

If you have made multiple transcriptions using different AI models, then the Copy all option will copy all those transcribed texts to the clipboard. This will also include any notes you have added, and the URL of the web page the image was found on.