Hi. If anyone having this trouble i will write my solution. I basically tried the solution in this thread: https://discussions.apple.com/thread/4712922
But i failed to delete the youtube.com lines in the mentioned hosts file because the system does not allow any typing changes in the file. But I found out that I have the same file out in downloads file of mine, i deleted the lines in there and replace it with the system's hosts file, which is allowed by computer. And now the problem is solved.
I'm having no issue playing YouTube videos in Safari 5.7.1 or Firefox 18 on OS X 10.6.8
Flash is currently at verison: 11.5.502.146
You have some sort of other problem and the /etc/hosts file shouldn't be messed with.
This is what it should look like and that's it. Anything esle is likely the result of malware or tinkering.
# Host Database
# localhost is used to configure the loopback interface
# when the system is booting. Do not change this entry.
Run through this list of fixes to repair your machine or have a local professional take a look at it.