What properties of lead nitrate cause this decomposition?

Lead(II) nitrate is a metal nitrate and has relatively weak bonds between the nitrate and metal at elevated temperature; on heating it breaks down because the nitrate is thermally unstable for that metal under those conditions. 

The presence of the nitrate group (NO₃⁻) means that when you apply heat the compound can produce NO₂ and O₂ while the metal oxidizes (Pb²⁺ → PbO). In other words its chemical composition and the instability at high temperature of the nitrate cause the decomposition. Also, the fact that lead(II) oxide is thermodynamically more stable than the original nitrate at high temperature makes the reaction favourable.