Ok, so this may be the most important assumption made by scientist. It is pretty simple to grasp, but it is also the linchpin to our understanding of the universe, and the way it works. We can use sports as an example that can help you relate to the importance of Universality. Take football. The Miami Dolphins can expect to play by the same rules whether they are in Florida, Texas, New Jersey, or even California. There is a standardization, and a uniformity to the rules that make up football, as well as other games like chess, tennis, or golf. Imagine what would happen if sporting rules changed based on the state you played in? Chaos would ensue. No team would want to play away from home because they would not know what to expect. Maybe a forward pass is a foul in Denver, and a Field Goal is worth 1 point in Seattle. Because no one would know what to expect, and there would be no way for teams to prepare for arbitrary changes in the rules, the games would also be unwatchable. The universality of the rules in football makes for a stable sporting environment where even competition can take place.
The same is the case with the universe. We assume that the laws of physics we observe are not unique to our neck of the woods. We assume that gravity works the same here, as it does in the Andromeda galaxy, and beyond. That electromagnetic waves have the same properties, and produce the same effects at home as they would in the far reaches of outer space.
How important is this? Well, if it were not true, we would not be able to discuss the physics of astronomy anymore. It would be a dead subject since you could not explain nor predict anything outside our small little corner of the universe.
Why do we feel confident that our assumption is right? Well, everywhere we point our telescopes, we see stars, solar systems, and galaxies that all play by the same rules. We have a flood of empirical evidence to support this most basic, and important assumption.