A simple space-time diagram, with time going upwards and distance to the right. Two "world lines" show the growing separation of our galaxy (vertical black line) and another galaxy (dashed red line). The shape of the red line is the classic a(t), showing deceleration followed by acceleration.
The light we see today, time t0, arrives when the galaxy is at distance r0. But this light set out at time tem when the galaxy was at a distance rem. As the equation suggests, the stretch of light waves mirrors, exactly, the cosmic stretch, and hence the redshift is related to the ratio r0/rem.