I would try to clear the app's cache. Go to your phones settings --> apps --> locate Hangouts --> clear cache/data. Then try and relaunch the app as normal.
(please note, this process listed might differ slightly on each phone - you want to go into your phones main Android settings and look for Apps - from there, locate Hangouts and then get to the storage or data portion to clear data/cache)
If that's a no-go, I would uninstall the app completely from the Play Store, restart the phone entirely, then reinstall the app and try to open at that point.
Report back if you have no luck!