Diagnosis
The TV has limited internal storage (around 5GB usable), and cached app data fills it up quickly, especially with multiple streaming apps. When storage is full, apps crash.
How to Fix It
Go to Settings > Applications > Manage Installed Applications. Select each app (especially the ones crashing) and choose 'Clear Cache'. Do NOT select 'Clear Data' unless you're okay with losing login info. Then, go to Settings > My Fire TV > Restart to reboot the TV.
If That Didn't Work
You need to uninstall apps you don't use. You can't move apps to external storage on this model. Freeing up space is the only fix.
Heads Up
This is a chronic issue with this model. Regularly clearing app caches (every few weeks) is part of the maintenance if you use a lot of streaming apps.