Quantum technologies fundamentally rely on quantum control, measurement, and feedback; however, a general understanding of many-body quantum dynamics under these conditions remains in its early stages. Such studies may provide insight into the dynamics of quantum computers undergoing active quantum error correction while running nontrivial quantum algorithms, as well as point to a more general understanding of the transition from quantum to classical physics in many-body systems. Measurement-induced transitions are a recently uncovered class of critical phenomena that generically occur when many-body unitary dynamics are interspersed with measurements at a tunable rate. As such, they realize a simple paradigm to begin investigating these questions from the viewpoint of statistical physics. We uncover precise connections between this phase transition and quantum error correction thresholds in the quantum channel capacity of open system dynamics. We then show how to define a local order parameter for the transition that is defined in terms of the ability of the system to store one bit of quantum information. Using this order parameter, we identify scalable probes of the transition that are immediately applicable to advanced quantum computing platforms such as trapped ions, superconducting qubits, or neutral atoms. Studying this class of measurement-driven many-body dynamics may potentially lead to more efficient realizations of scalable, fault-tolerant quantum computing.