  Extreme emission line galaxies (EELGs) exhibit large equivalent widths (EW)
in their rest-optical emission lines ([OIII]$\lambda5007$ or H$\alpha$
rest-frame EW$ > 750\r{A}$) which can be tied to a recent upturn in star
formation rate, due to the sensitivity of the nebular line emission and the
rest-optical continuum to young ($<10$Myr) and evolved stellar populations,
respectively. By studying a sample of 85 star forming galaxies (SFGs), spanning
the redshift and magnitude interval $3 <z<9.5$ and $-16>$ M$_{UV}>-21$, in the
JWST Advanced Deep Extragalactic Survey (JADES) with NIRSpec/prism
spectroscopy, we determine that SFGs initiate an EELG phase when entering a
significant burst of star formation, with the highest EWs observed in EELGs
with the youngest mass-weighted ages ($<5$ Myr old) and the highest burst
intensity (those with the highest proportion of their total stellar mass formed
in the last 10 Myr). We spectroscopically confirm that a greater proportion of
SFGs are in an EELG phase at high redshift in our UV-selected sample
($61\pm4\%$ in our $z>5.7$ high-redshift bin, compared to $23^{+4}_{-1}\%$ in
our lowest-redshift bin $3<z<4.1$) due to the combined evolution of
metallicity, ionisation parameter and star formation histories (SFH) with
redshift. We report that the EELGs within our sample exhibit a higher average
ionisation efficiency ($\log_{10}(\xi_{ion}^{HII}/$erg$^{-1}$Hz)$=25.5\pm0.2$)
than the non-EELGs. High-redshift EELGs therefore comprise a population of
efficient ionising photon producers. Additionally, we report that $53\%$ (9/17)
of EELGs at $z>5.7$ have observed Lyman-$\alpha$ emission, potentially lying
within large ionised regions. The high detection rate of Lyman-$\alpha$
emitters in our EELG selection suggests that the physical conditions associated
with entering an EELG phase also promote the escape of Lyman-$\alpha$ photons.
