The history of the universe is recorded in the stars. Seriously. By looking into space, we can see events that occurred billions of years ago. Let me explain.
The sun is 8 light minutes from the earth. This means it is far enough away that light from the sun does not reach the earth for 8 minutes. If the sun were to suddenly explode, we wouldn’t even know it until 8 minutes later. By looking at the sun, we are looking 8 minutes into history.
The distances to about 100,000 of the closest stars to earth has been directly measured using basic trigonometry. These stars are distances of 4 to 1600 light years from the earth and represent less than a ten-thousandth of a percent of the stars in our galaxy. Studying these stars reveals a definite, reliable connection between color, brightness, and distance. Because the color and brightness can be observed for stars that are farther away, this connection can be used to calculate the distances to other stars in our galaxy. It turns out that some of the stars visible in the night sky (such as V762 Cas) are up to about 15,000 light years away.
But that is just in our galaxy. There are many other galaxies that are close enough for astronomers to be able to distinguish individual stars. Using the same method, they have determined the distances to these galaxies (such as NGC 7320, at 40 million light years). And this represents only a small percentage of all the galaxies out there. Most of them are much farther than that.
So when we look at a galaxy that is 40 million light years from earth, we see it as it was 40 million years ago. This presents a problem for the young-earth model. How can we see a galaxy as it was 40 million years ago if it didn’t exist 40 million years ago?
I have often heard it suggested that God could have made the light in transit so that we could see it immediately. However, this explanation has a very big problem. We do not just see constant streams of light, but we see events being communicated in that light. For example, we have seen stars exploding (such as SN 2008D). If we observe the death of a star in a beam of light created in transit, then we are seeing a star that never really existed, and God had encoded in a beam of light the history of an event that never actually took place.
Over the course of many years, various people have attempted to create scientific models that can explain how we can see these stars without requiring the belief in a universe that is billions of years old. To date, nobody has succeeded in creating such a model that is compatible with the observed properties of the universe.1 In fact, the very proposition of a young universe that obeys general relativity and in which distant starlight is visible can mathematically be proven impossible.2
However, the inflationary big bang model does a very good job at explaining the observed properties of the universe. It is consistent both with general relativity (which has itself withstood rigorous testing for almost a century) and with observational data.3
So what is the most obvious explanation for light being visible from stars billions of light years away? The universe is billions of years old.
 Perhaps the most well-known model in the young-earth creationist community is one that has been proposed by Dr. Russell Humphreys and that purports to solve the distant starlight problem in the context of a young universe. However, this model fails on a number of counts, which include misapplication of known laws of physics and lack of conformity with observed properties of the universe. See reference below for details.
 Ross, H. (1999). The Unraveling of Starlight and Time. Retrieved from http://www.reasons.org/articles/the-unraveling-of-starlight-and-time
 Although there is one discrepancy (the horizon problem) between observation and the original formulation of the big bang model, this problem has been solved by the inclusion of an initial period of rapid expansion (inflation) in the model. This inflationary hypothesis has recently been validated by the discovery of the gravitational waves that it predicted.