This just happened in the past week or so. The videos will play fine using Chrome, but not Safari. Did something happen at youtube? Is this a flash issue? Something with Safari?
Most YouTube content requires the Flash plugin.
Uninstall your current copy of Flash then reinstall new > Troubleshoot Flash Player | Mac OS
Then quit and relaunch Safari.
From the Safari menu bar click Safari > Empty Cache
Restart your Mac, try a video.
If you continue to have problems, go to ~/Library/Preferences/com.adobe.flashplayer.installmanager
Move the Cache.db file from the com.adobe.flashplayer.installmanager folder to the Trash.
Quit then relaunch Safari, try a video.
Safari extensions can prevent some video from streaming. Go to Safari > Preferences then select select the Extensions tab.
If you have any installed, switch to OFF, quit then relaunch Safari, try again.
If you have ClickToFlash installed in ~/Library/Internet Plug-Ins, move that plug in to the Desktop. Quit then relaunch Safari to test.