Open System Preferences > Flash Player then select the Advanced tab.
Click Delete All under Browsing Data and Settings
Not empty the Safari cache.
From your Safari menu bar click Safari > Preferences then select the Advanced tab.
Select: Show Develop menu in menu bar
Now click Develop from the menu bar. From the drop down menu click Empty Caches.
Now try a video. If that doesn't help, troubleshoot Safari extensions and third party plugins.
From the Safari menu bar click Safari > Preferences then select the Extensions tab. Turn that OFF, quit and relaunch Safari to test. If that helped, turn one extension on then quit and relaunch Safari to test until you find the incompatible extension then click uninstall.
If it's not an extensions issue, try troubleshooting third party plug-ins.
Back to Safari > Preferences. This time select the Security tab. Deselect: Allow all other plug-ins. Quit and relaunch Safari to test.
If that made a difference, instructions for troubleshooting plugins here.