If you get this page, specifically the 'Search Blocked' page, it is because you do not have safe search enabled. First I will show you how to enable it, then I will proceed to explain the policy.
To enable on google:
While logged in to google, navigate to https://www.google.com/preferences
You should get this page.
The very fist setting is Safe Search filters. Simply click the checkbox that says 'Turn on SafeSearch'. Now scroll to the very bottom of the page, where you will find the blue 'Save' button. Click save.
Navigate to youtube's homescreen. If you are not logged in, go ahead and log in now.
Select your profile in the top right.
As you can see here I am a fan of movie trailers, computer science, Susanne Sundfør, and Meshuggah. You will also see the restricted mode setting appear in the corresponding dropdown.
You will now see the restricted mode setting will now show up. Simply click the switch to turn it on.
Now you are using safe search!
PART 2: WHY?
Our firewall uses a setting called safe search enforcement. This is tied into a technology called SSL decryption/inspection. When your computer connects to a website, it usually uses SSL which encrypts all the traffic to and from the website, so bad guys can't see the information. It also means our firewall can't see that information, and the firewall needs visibility into traffic to make sure nothing bad or compromising is going on. So the firewall intercepts the traffic, has a look, then re-encrypts it and sends it to the web server. This works ok on most things, but due to the nature of search engines it doesn't. We get traffic back from google, but google is the one talking to the various websites that it returns as search results. And because of this, alot of that traffic is encrypted and not readable by our firewall.
To address this, most search providers have a safe search option, where they run their own filter on the results when a clients firewall cannot. Our firewall knows when this is turned on or off, and blocks search results if they have not been run through a safe search filter. While this presents a small inconvenience, ultimately it prevents search results that you don't want showing up from showing up. You may search something completely innocuous, and have explicit results because someone tagged their video wrong, or gave a misleading title to their image. Safe Search is there to prevent this, so you don't search something that happens to be a double entedre, and end up showing your students something they should not see :P