Welcome to Android Central! Does this happen on both wi-fi and mobile data?
Try uninstalling FB Messenger, then make sure your main Facebook app is fully updated. Then try installing Messenger again.
If that doesn't work, then try going to Settings>Apps>All, select Google Play Store, and Clear Cache/Clear Data, then Force Stop. Do the same for Download Manager. Now try again.
If that doesn't work, try going to Settings>Accounts-Google, select your Google account, and remove it completely from the phone. Then add it back again, and see if that helps.