We report an angle-resolved photoemission spectroscopy (ARPES) study of KFe$_2$As$_2$ and CsFe$_2$As$_2$, revealing the existence of a van Hove singularity affecting the electronic properties. As a result of chemical pressure, we find a stronger three-dimensionality in KFe$_2$As$_2$ than in CsFe$_2$As$_2$, notably for the 3$d_{z^2}$ states responsible for the small three-dimensional hole-like Fermi surface pocket reported by quantum oscillations. Supported by first-principles calculations, our ARPES study shows that the van Hove singularity previously reported in KFe$_2$As$_2$ moves closer to the Fermi level under negative chemical pressure. This observation, which suggests that the large density-of-states accompanying the van Hove singularity contributes to the large Sommerfeld coefficient reported for the AFe$_2$As$_2$ (A = K, Rb, Cs) series, is also consistent with the evolution of the inelastic scattering revealed by transport under external pressure, thus offering a possible interpretation for the origin of the apparent change in the superconducting order parameter under pressure. We find that the coherent spectral weight decreases exponentially upon increasing temperature with a characteristic temperature $T^*$. We speculate how the low-energy location of the van Hove singularity and the presence of a low-energy peak in the phonon density-of-states can relate to the high-temperature crossover observed in various electronic and thermodynamic quantities.
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