There are three possible solutions that have been reported to work.
The easiest but least likely - reboot.
Or right-click or CONTROL-click over Audacity.app, click “Show Package Contents”, expand the “Contents” and “MacOS” folders then double-click the “Audacity” shell application. The Terminal will open then Audacity will launch.
Or it’s possible that files in the Audacity app have the xattribute “xattr com.apple.quarantine” set. This means that Gatekeeper blocks them. To fix it, open a Terminal (Applications > Utilities : Terminal.app) then type or paste:
sudo xattr -r -d com.apple.quarantine /Applications/Audacity/
and hit ENTER on your keyboard. Type your admin password (you won’t see it on the screen) then hit ENTER again.
Please let us know if this helps (and the answer to Steve’s question about whether this is a first installation of Audacity). Also is Sierra an upgrade over a previous version of Mac OS X?
Gale