  Recent observation of $\sim$ 10 times higher critical temperature in FeSe
monolayer compared with its bulk phase has drawn a great deal of attention
because the electronic structure in the monolayer phase appears to be different
than bulk FeSe. Using a combination of density functional theory and dynamical
mean field theory, we find electronic correlations have important effects on
the predicted atomic-scale geometry and the electronic structure of the
monolayer FeSe on SrTiO$_3$. The electronic correlations are dominantly
controlled by the Se-Fe-Se angle either in the bulk phase or the monolayer
phase. But the angle sensitivity increases and the orbital differentiation
decreases in the monolayer phase compared to the bulk phase. The correlations
are more dependent on Hund's J than Hubbard U. The observed orbital selective
incoherence to coherence crossover with temperature confirms the Hund's
metallic nature of the monolayer FeSe. We also find electron doping by oxygen
vacancies in SrTiO$_3$ increases the correlation strength, especially in the
$d_{xy}$ orbital by reducing the Se-Fe-Se angle.
