In my stats I can see that I have "unlocked a ward-skin". I want to activate this skin, but I don't know where or how to do so. I asked some friends and they said I have to go to the shop, but I can't find this skin there.
You havent technically unlocked the ward skin yet.
The skin you have "Unlocked" will be rewarded with the end of season 5. You'll find all the other ward skins the following way:
In the Client shop you'll find a tab on the left side that's labeled "Ward Skins". If you click on it you should see a graphical display of all the ward skins. There's also a filter function to display the different kind of skins (Teamranked rewards are under "Limited availability"). To switch back to the default skin you can click on the "Use default wards" icon in the top right corner.
In the shop, on the left side, you'll see a
Ward Skins option, click on it and from there, you'll see your owned skin with a checkbox on the upper left corner which you check to select your desired skin.
Ward skins on this solomid page to see this :