A person who is batshit crazy is certifiably nuts. The phrase has origins in the old fashioned term "bats in the belfry." Old churches had a structure at the top called a belfry, which housed the bells. Bats are extremely sensitive to sound and would never inhabit a belfry of an active church where the bell was rung frequently. Occasionally, when a church was abandoned and many years passed without the bell being rung, bats would eventually come and inhabit the belfry. So, when somebody said that an individual had "bats in the belfry" it meant that there was "nothing going on upstairs" (as in that person's brain). To be BATSHIT CRAZY is to take this even a step further. A person who is batshit crazy is so nuts that not only is their belfry full of bats, but so many bats have been there for so long that the belfry is coated in batshit. Hence, the craziest of crazy people are BATSHIT CRAZY.
Dude that guy on the corner wears a tinfoil hat and ripped all the wires out of his house so the government couldn't listen to his thoughts.