There are many reasons why depletion of natural vegetation may cause depletion of wildlife.
For many animals, vegetation is a primary, if not only, source of nutrition. For plant-eating animals, a decline in the availability of vegetation means a decline in food needed to sustain life. If plant life decreases due to drought, climate change, other natural disaster, or for some man-made reason such as pollution or land being developed for human use, the effect on the wildlife of the area is the same - less to eat leads to fewer animals surviving.
Natural vegetation serves as habitat and/or camouflage for some animals. Wildlife will migrate away from areas that don't offer protection from enemies or places to nest and raise offspring, leading to depletion of wildlife in a given location.