When you login to your farm, you'll likely see the Mystery Game pop-up automatically, telling you that you have a free dart. If not, you can go into the store and click on the Mystery Game manually as though you were going to pay to play. There, you should see that you have a Free Play available, in place of the button that would normally charge your Farm Cash. If this is anything like previous free Darts, you can hold onto it for future use, if this week's items don't do it for you.
For your reference, this week offers the following prizes:
Remember, the prize you'll end up with is random. For instance, I won the Dragon Fountain with my free dart, but you may have better luck and walk away with the Nightmare Pegasus (you lucky farmer, you)! Either way, make sure to claim your free dart as soon as possible, even if you don't use it right away, as it's likely it won't be available forever.
Did you receive a free Mystery Game dart when logging into FarmVille? If you've already thrown it at the board, which item did you end up winning? Sound off in the comments.