The answer to 'why is my mosquito droppings' is that you are likely seeing mosquito eggs or larvae, not actual droppings. Mosquitoes lay their eggs in standing water, and the small, wriggling larvae that hatch from those eggs may appear to be 'droppings' to the untrained eye.
The most common places to find mosquito eggs and larvae are in birdbaths, gutters, pet water bowls, and other areas where water collects. To get rid of mosquitoes, you'll need to eliminate any standing water around your home and treat any areas with mosquito larvae using an insecticide or biological control like mosquito dunks.