We present the results from a VLT/SINFONI and Keck/NIRSPEC near-infrared spectroscopic survey of 16 Lyman-alpha emitters (LAEs) at $z$ = 2.1 - 2.5 in the COSMOS and GOODS-N fields discovered from the HETDEX Pilot Survey. We detect rest-frame optical nebular lines (H$alpha$ and/or [OIII]$lambda$5007) for 10 of the LAEs and measure physical properties, including the star formation rate (SFR), gas-phase metallicity, gas-mass fraction, and Ly$alpha$ velocity offset. We find that LAEs may lie below the mass-metallicity relation for continuum-selected star-forming galaxies at the same redshift. The LAEs all show velocity shifts of Ly$alpha$ relative to the systemic redshift ranging between +85 and +296 km s$^{-1}$ with a mean of +180 km s$^{-1}$. This value is smaller than measured for continuum-selected star-forming galaxies at similar redshifts. The Ly$alpha$ velocity offsets show a moderate correlation with the measured star formation rate (2.5$sigma$), but no significant correlations are seen with the SFR surface density, specific SFR, stellar mass, or dynamical mass ($lesssim$ 1.5$sigma$). Exploring the role of dust, kinematics of the interstellar medium (ISM), and geometry on the escape of Ly$alpha$ photons, we find no signature of selective quenching of resonantly scattered Ly$alpha$ photons. However, we also find no evidence that a clumpy ISM is enhancing the Ly$alpha$ equivalent width. Our results suggest that the low metallicity in LAEs may be responsible for yielding an environment with a low neutral hydrogen column density as well as less dust, easing the escape of Ly$alpha$ photons over that in continuum-selected star-forming galaxies.
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