How do scientists determine the rate at which the universe is expanding? They charted the distances and speeds of extremely remote galaxies by studying their exploding stars, or supernovae.
Stars explode fairly often. In a galaxy like ours, about three stars explode every century. So if you observe enough galaxies, you can spot a new supernova every few nights.
These supernovae are very bright. They also have a uniform brightness, so scientists could use an observed supernova's brightness to determine its distance. It's similar to car headlights on a highway: the farther away the car, the dimmer its headlights.
They also measured the supernova's light over time to determine how fast it was moving away from us. By analyzing the data from hundreds of supernovae, they could measure how the universe's rate of expansion has changed over time.