As the universe expands, light crosses the expanding space (always moving locally at
speed c). The stretch in wavelength is the same as the cosmic stretch: 
now / emit = 1 + z = dnow / dem. You can think of the wave
being stretched as it crosses the expanding space.
Diagram 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.