As the universe expands, light crosses the expanding grid (always moving locally at speed c). The redshift is simply R(tobs) / R(tem), the ratio of the scale factors between observing and emission. You can think of the wave being stretched as it crosses the expanding grid.
A similar diagram taken from Seed's astr-101 text. Notice that from top to bottom not only does the grid expand, but the area shown pans from right to left.
Figure from OU text "Galaxies and Cosmology"