Install by downloading the latest release – if you use Firefox, make sure to right-click and save the XPI file, not just clicking it – and then in Zotero:
After the initial installation, the plugin will auto-update to newer releases, so you should need to perform the process described here only once.
Note that the default setting of BBT will generate different citekeys than Zotero would itself generate; the keys from Zotero are not always safe for use in bibtex/biber. If you want to get the stock zotero keys, set the pattern in the preferences to
[zotero]. I very much recommend not choosing
[zotero] as your pattern unless you have existing articles that use keys generated by previous Zotero-native BibTeX export.
Note that BBT needs to be installed in Zotero, not Firefox. There used to be two versions of Zotero, one standalone, and one that was installed in Firefox. BBT is a Zotero extension and lives where Zotero lives; Zotero is no longer available as a Firefox plugin, only as standalone.
On first install you will be asked a few questions. Please read these instead of just clicking through.