Kondo insulating materials lie outside the usual dichotomy of weakly versus correlated-band versus Mott-insulators. They are metallic at high temperatures but resemble band insulators at low temperatures because of the opening of an interaction-induced band gap. The first discovered Kondo insulator (KI) SmB$\mathrm{<mrow\; data-mjx-texclass="ORD">6</mrow>}$ has been predicted to form a topological KI (TKI). However, since its discovery thermodynamic and transport anomalies have been observed that have defied a theoretical explanation. Enigmatic signatures of collective modes inside the charge gap are seen in specific heat, thermal transport, and quantum oscillation experiments in strong magnetic fields. Here, we show that TKIs are susceptible to the formation of excitons and magnetoexcitons. These charge neutral composite particles can account for long-standing anomalies in SmB$\mathrm{<mrow\; data-mjx-texclass="ORD">6</mrow>}$.