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The ability to control electronic states at the nanoscale has contributed to our modern understanding of condensed matter. In particular, quantum dot circuits represent model systems for the study of strong electronic correlations, epitomized by the Kondo effect(1-3). We use circuit quantum electrodynamics architectures to study the internal degrees of freedom of this many-body phenomenon. Specifically, we couple a quantum dot to a highquality- factor microwave cavity to measure with exceptional sensitivity the dot's electronic compressibility, that is, its ability to accommodate charges. Because electronic compressibility corresponds solely to the charge response of the electronic system, it is not equivalent to the conductance, which generally involves other degrees of freedom such as spin. Here, by performing dual conductance and compressibility measurements in the Kondo regime, we uncover directly the charge dynamics of this peculiar mechanism of electron transfer. The Kondo resonance, visible in transport measurements, is found to be ` transparent' to microwave photons trapped in the high-quality cavity, thereby revealing that (in such a many-body resonance) finite conduction is achieved from a charge frozen by Coulomb interaction. This freezing of charge dynamics(4-6) is in contrast to the physics of a free electron gas. We anticipate that the tools of cavity quantum electrodynamics could be used in other types of mesoscopic circuits with many-body correlations7,8, providing a model system in which to perform quantum simulation of fermion-boson problems.