The metal-organic-framework (MOF) compound Cu$_3$(HOTP)$_2$ is a small-gap semiconductor containing a kagome lattice of antiferromagnetically coupled $S$=1/2 Cu$^\mathrm{II}$ spins with exchange coupling of order 2 K. First principles spin-polarized DFT+U electronic structure calculations obtain an in-plane nearest-neighbour exchange coupling that matches experiment. Muon spin relaxation confirms no magnetic ordering down to 50~mK and sees spin fluctuations diffusing on a 2D lattice, consistent with a quantum spin liquid (QSL) ground state. Reduction of the diffusion rate on cooling from the paramagnetic region to the low temperature QSL region is assigned to the effects of quantum entanglement. Combined analysis of the spin diffusion, magnetic susceptibility and specific heat in the QSL region suggests close proximity to a quantum critical point and a large density of low energy spin-less electronic excitations. A Z$_2$-linear Dirac model for the spin excitations of the QSL is found to provide the best overall consistency with experiment.