We use the speed of light to calculate how far apart the stars are, and to estimate the size and age of the universe. Speed-of-light calculations are what confirmed that the universe is expanding. And because light takes a certain amount of time to get here, when we look at the stars, we're literally looking back in time, to what the universe was like millions of years ago:
"The speed of light is particularly important in astronomy. Due to the vast distances involved it can take a very long time for light to travel from its source to Earth. For example, it takes 13 billion years for light to travel to Earth from the faraway galaxies viewed in the Hubble Ultra Deep Field images. Those photographs, taken today, capture images of the galaxies as they appeared 13 billion years ago (near the beginning of the universe)."