Overleaf V2 - How to get BBL File?

You can get the bbl file if you click on the "log and other files" icon next to compile, recompile icon just above the pdf viewer. Then you can scroll down, and at the bottom you have a "log and other file" icon/button, you can select it to download the bbl file.

(Disclaimer: I'm a support personnel at Overleaf.)

Click on "Submit" 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.