According to this blog post from January 21, 2016 by Chris Mills (a senior tech writer at Mozilla) you can't use it yet in web environment on FF ...
... because the UX/UI to allow users to grant an app permission to use it is not yet implemented.
...
Here is a bug report for which you can vote.
And currently, you'll have to be in the chrome
environment (extension) to use it.
Update winter 2019
The linked bug report has been marked as fixed.
To enable the SpeechRecognition in Firefox Nightly > 72, go to about:config
and switch the flags media.webspeech.recognition.enable
and media.webspeech.recognition.force_enable
to true.
However note this is an online service, which means you'll need an internet connection.