The word fruit can be very specific, referring to foods like berries and apples, or a bit more general to refer to how an organism reproduces. So tomatoes and rosehips are the fruits of those plants; they contain the seeds. Likewise, mushrooms are the fruits of a fungus; they contain spores. So when a fungus starts to produce mushrooms, we call that fruiting.
Specifically, mushrooms are the reproductive structure of the fungus. The block in your kit is primarily comprised of mycelium (the vegetative body of a fungus), and it's food (a bag full of supplemented sawdust). As the mycelium grows, consuming the sawdust mix, it uses up the nutrients available in the bag. When it's fully established, has run out of nutrients or experiences other environmental cues like a temperature or oxygen shift, that's a cue for the fungus to fruit and produce mushrooms so that it can move on to another food source.