We use a custom shear cell coupled to an optical microscope to investigate at the particle level the yielding transition in concentrated emulsions subjected to an oscillatory shear deformation. By performing experiments lasting thousands of cycles on samples at several volume fractions and for a variety of applied strain amplitudes, we obtain a comprehensive, microscopic picture of the yielding transition. We find that irreversible particle motion sharply increases beyond a volume-fraction dependent critical strain, which is found to be in close agreement with the strain beyond which the stress-strain relation probed in rheology experiments significantly departs from linearity. The shear-induced dynamics are very heterogenous: quiescent particles coexist with two distinct populations of mobile and 'supermobile' particles. Dynamic activity exhibits spatial and temporal correlations, with rearrangements events organized in bursts of motion affecting localized regions of the sample. Analogies with other sheared soft materials and with recent work on the transition to irreversibility in sheared complex fluids are briefly discussed.