  Galaxies in the reionization-era have been shown to have prominent
[OIII]+H$\beta$ emission. Little is known about the gas conditions and
radiation field of this population, making it challenging to interpret the
spectra emerging at $z\gtrsim6$. Motivated by this shortcoming, we have
initiated a large MMT spectroscopic survey identifying rest-frame optical
emission lines in 227 intense [OIII] emitting galaxies at $1.3<z<2.4$. This
sample complements the MOSDEF and KBSS surveys, extending to much lower stellar
masses ($10^7-10^8 M_\odot$) and larger specific star formation rates ($5-300$
Gyr$^{-1}$), providing a window on galaxies directly following a burst or
recent upturn in star formation. The hydrogen ionizing production efficiency
($\xi_{\rm{ion}}$) is found to increase with the [OIII] EW, in a manner similar
to that found in local galaxies by Chevallard et al. (2018). We describe how
this relationship helps explain the anomalous success rate in identifying
Ly$\alpha$ emission in $z\gtrsim7$ galaxies with strong [OIII]+H$\beta$
emission. We probe the impact of the intense radiation field on the ISM using
O32 and Ne3O2, two ionization-sensitive indices. Both are found to scale with
the [OIII] EW, revealing extreme ionization conditions not commonly seen in
older and more massive galaxies. In the most intense line emitters, the indices
have very large average values (O32 $=9.1$, Ne3O2 $=0.5$) that have been shown
to be linked to ionizing photon escape. We discuss implications for the nature
of galaxies most likely to have O32 values associated with significant LyC
escape. Finally we consider the optimal strategy for JWST spectroscopic
investigations of galaxies at $z\gtrsim10$ where the strongest rest-frame
optical lines are no longer visible with NIRSpec.
