Beam-recoil transferred polarizations for the exclusive e→p→e'K+Λ→,Σ→0 reactions have been measured using the Continuous Electron Beam Accelerator Facility's large acceptance spectrometer (CLAS) at the Thomas Jefferson National Accelerator Facility. New measurements have been completed at beam energies of 4.261 and 5.754 GeV that span a range of momentum transfer Q2 from 0.7 to 5.4 GeV2, invariant energy W from 1.6 to 2.6 GeV, and the full center-of-mass angular range of the K+ meson. These new data add to the existing CLAS K+Λ measurements at 2.567 GeV, and provide the first-ever data for the K+Σ0 channel in electroproduction. Comparisons of the data with several theoretical models are used to study the sensitivity to s-channel resonance contributions and the underlying reaction mechanism. Interpretations within two semiclassical partonic models are made to probe the underlying reaction mechanism and the ss̄ quark-pair creation dynamics.