Deep VLA 1.4 GHz radio continuum imaging of Abell 2255 is presented. This cluster is among the better nearby candidates for rich cluster-cluster merger systems, with evidence including an elongated X-ray morphology, the presence of a radio halo, and substructure present in its galaxy distribution. Our radio observations reach an rms sensitivity of ~40 μJy beam-1, enabling us to detect (at 5 σ) star formation rates as low as 1.4 M⊙ yr-1 from the center of the cluster out to a radial distance of 3 h Mpc. The radio data are complemented by optical imaging and a large spectroscopic database, allowing us to separate all galaxies with M ≤ -20 into cluster members and foreground/background galaxies. The spectra are also used to associate the galaxies' radio emission with either star formation or active galactic nuclei (AGNs). We compare the resulting cluster radio galaxy population with those of 19 other nearby Abell clusters and find strong evidence for an increase in the frequency of radio galaxies in Abell 2255. This increase is seen in two separate types of galaxies: powerful radio AGNs and optically faint star-forming galaxies. The optical spectra of the latter often indicate current or recent starbursts, and these galaxies appear to be distributed along an axis perpendicular to the probable merger axis. We assess these factors in light of models of galaxy evolution and suggest that the cluster-cluster merger is responsible for triggering galaxy activity in Abell 2255.