Dark energy has come up a number of times, and it's a good case study to show that we have looked out into the universe many times and found things that don't obey our current model of the universe. Rather than deciding that our physics is a special case, we revise our model of the universe ever finer and come up with physics that fits both what we observe locally and at a cosmic scale. That we can successfully make predictions with this ever refined model very strongly suggests that physics is the same everywhere.
Back in the 19th century evidence was mounting that the Earth was billions of years old. This presented many problems, not the least of which was how did the Sun manage to shine that long? No known fuel source at the time could sustain such a fire for billions of years. Even nuclear fission couldn't explain it.
The debate raged back and forth, and evidence mounted that yes, the Earth was billions of years old, so something is wrong with our understanding of the Sun. Eventually nuclear fusion was discovered and that resolved the paradox.
Fast forward to the 21st century and we have new problem with gravity at a galactic scale. We're observing that galaxies are accelerating away from each other rather than slowing down as we'd expect. We've given this accelerating force a name, "dark energy", even though we know very, very little about it.
Again, evidence is mounting that this is real. And, again, it's likely not that our local physics is different, but that dark energy is so weak it only has an effect on a galactic scale. Much like how gravity is so weak you don't notice its force until you get a few trillion kilograms of matter together; there's so many stronger forces interfering with your observations.
The density of dark energy is estimated to be roughly 7e−30 g/cm3. That's roughly the same as matter, but unlike matter which clumps up in galaxies, it is evenly distributed across the universe. And, unlike matter, it only interacts via gravity. Physics (probably definitely) works the same everywhere, but we have to look out into space (and back in time) to get observations on the necessary scale.
The other option, that our understanding of gravity is wrong, so far doesn't work. All of the alternative theories of gravity people have tried to explain dark energy fail to match other observations. Like trying to fit a wrong sized carpet into a room, fit one corner and the other pops out.
We could say "well I guess physics is just different in all those other places", but exceptions like that result in inelegant messes that, and this is very important, offer no predictive power. It's not just that it's inelegant, but it also results in a model of the universe that is less useful. Physicists measure how good their model of the universe is by how well it matches existing observations, but also how well it predicts future observations. "Gravity just works different around Somewheretarius 5" doesn't tell us anything about the rest of space, whereas dark energy as a cosmological constant neatly explains and predicts across the whole universe.
Another reason to believe that we're not in a local physics bubble is so many of our predictions about the universe at very large scales come out correct. This is the heart of science: come up with a theory about how the world works, make some predictions based on it, see if they turn out correct. If they do, your theory is strengthened! If they don't, back to the drawing board.
The two biggest examples in recent memory are the Cosmic Microwave Background (CMB) and Gravitational Waves. Both rely heavily on the assumption that physics is homogeneous across the entire universe.
A prediction was made both about what the CMB would look like, and what gravitational waves the merger of two black holes would make. WMAP's observations of the CMB and LIGO's observation of a black hole merger match those predictions extremely closely.
These results mean our physics is correct at the scale of the entire history of the universe and even for crazy things like black holes. We don't have to pepper our physical laws with exceptions and special cases, instead we refine them ever more, all from our little ball of mud.
The WMAP result is so spectacular that xkcd crowed "Science. It works, bitches." with a graph of the predicted and observed CMB black body radiation.