In bulk form the prototypical iron based superconductor FeSe has a humble \Tc of only 8~K. When grown on the ubiquitous perovskite SrTiO$_3$, however, the interface appears to exhibit an enormous superconducting enhancement of up to $\approx$60~K, in excess of any other known Fe based material in bulk. However, due to the challenges inherent to the study of air-sensitive interfacial superconductors, the nature and origin of this enhancement effect remain mysterious. A complete understanding of the underlying enhancement mechanism in this novel system could open the door to engineering new interfacial superconductors with higher~$T_c$'s. In this dissertation, we use the synergistic combination of molecular beam epitaxy (MBE), angle-resolved photoemission spectroscopy (ARPES), and \insitu resistivity to exhaustively explore the novel superconducting behavior of \FeSeSTO ultra-thin films. In doing so, we reveal an unprecedentedly large pseudogap regime in single unit cell thick (1-UC) \FeSeSTO where the spectroscopic gap persists at elevated temperatures \smash{(\Tgap $\approx$ 70 K)} well above the onset of zero resistance \smash{(\Tzero$\approx$ 30 K)}. Furthermore, by systematically comparing transport behavior across many samples, we observe distinct signatures of two-dimensional (2D) superconducting phase fluctuations, revealing a mixed vortex state hosting incoherent Cooper pairs which ultimately drives the pseudogap observed by ARPES at elevated temperatures. This naturally explains the unconventional gap behavior and reconciles many previously mysterious longstanding discrepancies between transport and spectroscopic behaviors in 1-UC \FeSeSTOns. Building on this result, we exploit the synthesis power of MBE to study how superconductivity at the FeSe/SrTiO$_3$ interface responds to the additional growth of extra layers, using \insitu transport measurements to gain access to the buried interface. In contrast to previous reports, we show that the transport \Tc of the interfacial monolayer is robust and largely unaffected by additional FeSe layers, and that the earlier reported reduction of \Tc in thicker films can be explained by parallel transport channels in the presence of a broad, 2D transition. This places important constraints on the mechanism of the \Tc enhancement. In multi-layer samples, we successfully measure a macroscopic zero resistance state induced via alkali surface-doping, and map out the entire doping-dependent superconducting dome in the absence of interfacial effects. Surprisingly, we find that the onset of zero resistance for optimally surface-doped samples occurs at ($\approx$ 28 K) - nearly identical to the best interfacial films ($\approx$ 30 K). Finally, we performed comprehensive photon energy dependent measurements of the enigmatic 1-UC \FeSeSTO replica band features using synchrotron ARPES. Studying the evolution of the replica band at $M$ with increasing photon energy, we find conclusive evidence that the interfacial replica band in \FeSeSTO originates from an intrinsic electron-phonon coupling effect. This confirms that interfacial phonon coupling at atomic interfaces can produce nontrivial modifications to the underlying bandstructure, a finding with widespread implications. Taken together, our results demonstrate the critical role that limited dimensionality plays in the emergence of superconductivity in FeSe/SrTiO$_3$, and in interfacial high-$T_c$ systems more generally. Based on this understanding, we suggest several promising pathways forward for further research exploiting interfacial enhancement mechanisms of superconductivity.