Despite the tremendous interest in halide perovskites in solar cells, the reason that the all-inorganic perovskite CsPbI$_3$ is unstable at room temperature remains mysterious. Here single-crystal X-ray diffraction and powder pair distribution function (PDF) measurements characterize bulk perovskite CsPbI$_3$ from 100 to 295 K to elucidate its thermodynamic instability... While Cs occupies a single site from 100 to 150 K, it splits between two sites from 175 to 295 K with the second site having a lower effective coordination number, which along with other structural parameters suggests that Cs rattles in its coordination polyhedron. PDF measurements reveal that on the length scale of the unit cell, the Pb-I octahedra concurrently become greatly distorted, with one of the I-Pb-I angles approaching 82{\deg} compared to the ideal 90{\deg}. The rattling of Cs, low number of Cs-I contacts, and high degree of octahedral distortion cause the instability of perovskite-phase CsPbI$_3$. These results provide detailed structural information that may help to engineer greater stability of CsPbI$_3$ and other all-inorganic perovskites for use in solar cells. (read more)