Cell function depends on tissue rigidity, which cells probe by applying and transmitting forces to their extracellular matrix, and then transducing them into biochemical signals. Here we show that in response to matrix rigidity and density, force transmission and transduction are explained by the mechanical properties of the actin-talin-integrin-fibronectin clutch. We demonstrate that force transmission is regulated by a dynamic clutch mechanism, which unveils its fundamental biphasic force/rigidity relationship on talin depletion. Force transduction is triggered by talin unfolding above a stiffness threshold. Below this threshold, integrins unbind and release force before talin can unfold. Above the threshold, talin unfolds and binds to vinculin, leading to adhesion growth and YAP nuclear translocation. Matrix density, myosin contractility, integrin ligation and talin mechanical stability differently and nonlinearly regulate both force transmission and the transduction threshold. In all cases, coupling of talin unfolding dynamics to a theoretical clutch model quantitatively predicts cell response.