The actin-based cytoskeleton is a polymer network that plays an essential role in cell biology. By self-organizing into various local architectures, the cytoskeleton performs physiological functions that allow the cell to physically interact with its environment. It is also an example of biological active matter, consuming chemical free energy at a local scale to produce directed motion and do mechanical work. While it is well-known that cytoskeletal free energy transduction occurs, it has been a challenge to say anything quantitative about this far-from-equilibrium process due to the difficulty of making the necessary experimental measurements. This lack of methodology to quantify cytoskeletal energetics significantly hinders our understanding of the self-organization process underlying the cytoskeleton's physiological functionality. To address this research gap, we develop in this thesis an explicit computational method to quantify chemical and mechanical free energy changes during simulated cytoskeletal self-organization using the software package MEDYAN (Mechanochemical Dynamics of Active Networks). We then apply this tool in several studies to advance our understanding of the self-organization process and its thermodynamic characteristics. For instance, we analyze the thermodynamic efficiency of mechanical stress generation and the network's time-dependent dissipation rates under a range of network conditions. We also investigate the recent experimentally discovered phenomenon of cytoskeletal avalanches, which we identify in simulation as anomalous mechanical dissipation events. Our analysis clarifies the phenomenology and underlying mechanism of these avalanche events, which we propose may play an important role in cellular information processing. The in silico method developed in this thesis provides a new perspective on cytoskeletal self-organization and may be extended to investigate other biological active matter systems.