Let's get this out of the way first: The simulation argument doesn't prove that we're living in a computer simulation. The argument is built atop assumptions. If one or more of those assumptions proves to be false, the argument is invalid.
Beyond that, the argument is unfalsifiable. A falsifiable theory is one that can be disproven in an experiment or observation. Science and the scientific method depend upon falsifiability. If there are no criteria under which a theory could be disproven, it's unfalsifiable and unscientific. For example, if I claimed that you're always being followed by a 2-foot-tall (0.6-meter-tall) mouse that's invisible, impossible to touch and makes no noise, that's unfalsifiable. There's no way to disprove my statement, which removes it from the realm of science.
The simulation argument falls into this category -- if we were to use the test method suggested by the three physicists, a negative result wouldn't necessarily mean we could state with authority that we aren't in a simulation. Perhaps the simulation would prevent us from discovering the truth. That's why the argument is philosophical rather than scientific. But for argument's sake, what would it mean to us if our universe were just a simulation?
If we never have any way of knowing, there's no reason anything would change. From our perspective, the universe would be as it always has been. But imagine that we find a way to prove beyond any doubt that we're inside a computer simulation.
The religious implications would be dramatic. We would have proof that there is some sort of creator. That creator may or may not resemble our religious icons. Any announcement that our universe is just a simulation would likely encounter skepticism and denial across a broad spectrum of people. The cultural and social implications are enormous.
From a practical, day-to-day perspective, things might not change that much. Even if everything we know and can know is a simulation, we still exist within that universe. We still eat, breathe, live and die. The conditions around us don't change whether we're in reality or some other reality's virtual world.
That could shift if we found some way to interact with the beings that created the simulation. It could mean that our world is similar to the one in the movie "The Matrix" -- by changing some code, we could end up drastically changing ourselves or our environment. Or it might mean they get bored with their simulation and shut the whole thing off.
Ultimately, there's no way for us to know right now if our universe is a simulation or not. But it sure makes you think, doesn't it?