A precise description of quantum gravity is provided by the Anti-de Sitter/Conformal Field Theory correspondence and formulated as a quantum field theory in one fewer spacetime dimensions. This duality has shed light on many deep puzzles in quantum gravity, but the emergence of local physics on the gravitational side remains rather mysterious. I will present recent work explaining how this emergence can naturally be understood as a manifestation of quantum error correction, an idea used to protect quantum information from decoherence. I will use this new interpretation to make precise the deep connection between entanglement and spacetime geometry, and explore the quantum nature of black holes. This talk should be quite accessible to beginning graduate students and assumes no background in gravity, field theory, or quantum error correction.