Firefox 43 requires all extensions to be signed. See https://wiki.mozilla.org/Add-ons/Extension_Signing

The Lastpass extension is signed, and they haven't yet figured out what caused it to fail verification for some people. Uninstalling Lastpass via the Add-ons Manager and reinstalling from AMO solves the problem. https://addons.mozilla.org/en-US/firefox/addon/lastpass-password-manager/

Also see https://blog.lastpass.com/2015/12/easily-enable-lastpass-in-firefox-43.html/

If you have the version that uses a plugin or binary components then you would need a 64 bit version of LastPass.

Does it work with a 32 bit Firefox version?

You would have to contact LastPass to check if they support 64 bit Firefox versions.