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: 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: 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.