We present spectroscopic properties of 22 Ly$\alpha$ emitters (LAEs) at $z = 5.5 - 6.6$ with Ly$\alpha$ luminosity $\mathrm{log}( L_{\mathrm{Ly}\alpha} \, [\mathrm{erg} \, \mathrm{s}^{-1}]) = 42.4 - 43.5 $, obtained using VLT/MUSE as part of the Middle Ages Galaxy Properties with Integral Field Spectroscopy (MAGPI) survey. Additionally, we incorporate broad-band photometric data from the Subaru Hyper Suprime-Cam (HSC) Wide layer for 17 LAEs in our sample. The HSC-y band magnitudes show that our LAEs are UV-bright, with rest-frame absolute UV magnitudes $ -19.74 \leq \mathrm{M}_{\mathrm{UV}} \leq -23.27$. We find that the Ly$\alpha$ line width increases with Ly$\alpha$ luminosity, and this trend becomes more prominent at $z \gt 6$ where Ly$\alpha$ lines become significantly broadened ($\gtrsim+260 \, \mathrm{km}\, \mathrm{s}^{-1}$) at luminosities $\mathrm{log}( L_{\mathrm{Ly}\alpha} \, [\mathrm{erg} \, \mathrm{s}^{-1}]) \gt 43 $. This broadening is consistent with previous studies, suggesting that these sources are located inside larger ionised bubbles. We observe a slightly elevated ionising photon production efficiency estimated for LAEs at $z \gt 6$, which indicates that younger galaxies could be producing more ionising photons per UV luminosity. A tentative anti-correlation between ionising photon production efficiency and Ly$\alpha$ rest-frame equivalent width is noticed, which could indicate a time delay between production and escape of ionising photon primarily due to supernovae activity. Furthermore, we find a positive correlation between radius of ionised regions and Ly$\alpha$ line width, which again suggests that large ionised bubbles are created around these LAEs, which are allowing them to self-shield from the scattering effects of the intergalactic medium (IGM). We also detect two very closely separated LAEs at $z = 6.046$ (projected spatial distance between the cores is 15.92 kpc). This is the LAE pair with the smallest separation ever discovered in the reionisation epoch. The size of their respective bubbles suggests that they likely sit inside a common large ionised region. Such a closely separated LAE pair increases the size of ionised bubble, potentially allowing a boosted transmission of Ly$\alpha$ through neutral IGM and also supports an accelerated reionisation scenario.