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The lead-free halide double perovskite class of materials offers a promising venue for resolving issues related to toxicity of Pb and long-term stability of the lead-containing halide perovskites. We present a first-principles study of the lattice vibrations in Cs$_2$AgBiBr$_6$ , the prototypical compound in this class, and show that the lattice dynamics of Cs$_2$AgBiBr$_6$ is highly anharmonic, largely in regards to tilting of AgBr$_6$ and BiBr$_6$ octahedra. Using an energy and temperature dependent phonon spectral function, we then show how the experimentally observed cubic-to-tetragonal phase transformation is caused by the collapse of a soft phonon branch. We finally reveal that the softness and anharmonicity of Cs$_2$AgBiBr$_6$ yield an ultra-low thermal conductivity, unexpected of high symmetry cubic structures.