Recent terahertz conductivity measurements observed low-power-law frequency dependence of optical conduction within the Mott gap of the kagome lattice spin-liquid candidate Herbertsmithite. We investigate mechanisms for this observed sub-gap conductivity for two possible scenarios in which the ground state is described by (1) a U(1) Dirac spin liquid with emergent fermionic spinons or (2) a nearly critical Z[subscript 2] spin liquid in the vicinity of a continuous quantum phase transition to magnetic order. We identify new mechanisms for optical absorption via magnetoelastic effects and spin-orbit coupling. In addition, for the Dirac spin liquid scenario, we establish an explicit microscopic origin for previously proposed absorption mechanisms based on slave-particle effective field theory descriptions.National Science Foundation (U.S.) (Grant DMR 1104498)National Science Foundation (U.S.) (DRM-1005434)Simons Foundation (Award 229736)