Click on the Submit button above the preview panel, and then choose e.g. "arXiv". There will be an option to download a .zip with the .bbl included. Once you click the button, it should show a "Please wait" message and take a while to build the zip. Then the button comes back, only this time it is a download link that will immediately download the zip archive.
You can also download the .bbl file separately. Clicking on the "Logs and output files" icon next to "Recompile", then scroll right down to the bottom. Then click on "Other logs & output files"; you can then download the .bbl file.
Under the Project menu, if you click the arrow next to 'Download Zip', you will see three options:
For official journal templates on Overleaf that have a customized
'Submit to Journal' option in the topbar (as shown in the image above for the OSA), if that submission modal contains a download zip option it will be set to automatically download the appropriate set of files required for that publisher.
If you are submitting to a journal from Overleaf and have any questions, please get in touch, we'll be happy to help if we can!