  The two-dimensional (2D) material Cr$_2$Ge$_2$Te$_6$ is a member of the class
of insulating van der Waals magnets. Here, using high resolution angle-resolved
photoemission spectroscopy in a detailed temperature dependence study, we
identify a clear response of the electronic structure to a dimensional
crossover in the form of two distinct temperature scales marking onsets of
modifications in the electronic structure. Specifically, we observe Te
$p$-orbital-dominated bands to undergo changes at the Curie transition
temperature T$_C$ while the Cr $d$-orbital-dominated bands begin evolving at a
higher temperature scale. Combined with neutron scattering, density functional
theory calculations, and Monte Carlo simulations, we find that the electronic
system can be consistently understood to respond sequentially to the distinct
temperatures at which in-plane and out-of-plane spin correlations exceed a
characteristic length scale. Our findings reveal the sensitivity of the
orbital-selective electronic structure for probing the dynamical evolution of
local moment correlations in vdW insulating magnets.
