  Beam-recoil transferred polarizations for the exclusive $\vec{e}p \to e'K^+
\vec{\Lambda},\vec{\Sigma}^0$ reactions have been measured using the CLAS
spectrometer at Jefferson Laboratory. New measurements have been completed at
beam energies of 4.261 and 5.754 GeV that span a range of momentum transfer
$Q^2$ from 0.7 to 5.4 GeV$^2$, 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 $K^+\Lambda$ measurements at 2.567 GeV, and provide the
first-ever data for the $K^+\Sigma^0$ 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 semi-classical partonic models are made to probe the
underlying reaction mechanism and the $s\bar{s}$ quark-pair creation dynamics.
