  The quantum magnet SrCu$_2($BO$_3)_2$ and its remarkably accurate theoretical
description, the spin-$1/2$ Shastry-Sutherland model, host a variety of
intriguing phenomena such as a dimer ground state with a nearly flat band of
triplon excitations, a series of magnetization plateaux, and a possible
pressure-induced deconfined quantum critical point. One open puzzle originating
from inelastic neutron scattering and Raman experiments is the anomalous
broadening of the triplon modes at relatively low temperatures compared to the
triplon gap $\Delta$. We demonstrate that the experimentally observed
broadening is captured by the Shastry-Sutherland model. To this end, we develop
a numerical simulation method based on matrix-product states to simulate
dynamical spectral functions at nonzero temperatures accurately. Perturbative
calculations identify the origin of this phenomenon as a small energy scale
compared to $\Delta$ between single triplon and bound triplon states at the
experimentally relevant model parameters.
