To perform this fix, open up Steam, go to Library, right click Counter Strike: Global Offensive, and then click "Properties":
After opening up the properties window, head to "Local Files" and click "Verify Integrity of Game Files":
Wait for this to complete, and then restart your computer.

After completing the above, your CS:GO should be working fine, and the "Matchmaking Failed - VAC was unable to verify your game session" error should be gone. If, however, it's still occurring, continue reading Fix #2 below.

Repairing the Steam service is recommended by Valve here. To do this, all you need to do is:

Close Steam. Do this by right clicking Steam in your System Tray (bottom right hand corner) and clicking "Exit":

Open your Start Menu, search for "Run" and open the file named "Run":
Enter the following into the text box and press "OK":

"C:\Program Files (x86)\Steam\bin\SteamService.exe" /repair

An empty terminal should then pop up. Wait for this to close and re-open Steam - your issue should be fixed.

If your error isn't fixed, continue reading for Fix #3.

if the two fixes outlined above haven't worked, your issue is likely one that won't be resolved by changing a setting or restarting your computer. Whilst we recommend you search the web to see if there is anything else that might work for you, chances are, you're going to save time by reinstalling CS:GO. Reinstalling the game will get the issue fixed quickly, and will save you having to try lots of different fixes (that probably won't work).

To reinstall CS:GO completely, go to your Library in Steam, right click "Counter-Strike: Global Offensive" and click "Uninstall":
On the window that pops up, click "Delete":
Then you'll need to reinstall CS:GO. You can do this at https://store.steampowered.com/app/730/CounterStrike_Global_Offensive/. After reinstalling CS:GO, the "Matchmaking Failed - VAC was unable to verify your game session" error should be fixed!