Why (or when) should I put it in the fridge?
In nature, the shiitake will fruit in the autumn (and spring) as it can detect wild fluctuation of seasonal temperature. They can sense when it’s time to procreate. Shiitake are very cold tolerant and in nature may need to live-out winter in a frozen log.
You would typically use the refrigerator, or an ice bath if it hasn’t fruited in a couple of weeks, and the weather has been warm.