Can someone that knows a bit more about this explain why we believe that this dark matter substance must exist and not simply that we have an incorrect model of gravitation?
Because such a minimal change to the model ("there's just some more stuff we can't see") fixes a number of problems in one fell swoop (off the top of my head, galactic rotation curves, observations of gravitational lensing, a cosmological model compatible with obervations of CMB fluctuations, structure formation, cosmic evolution).
Theories of modified gravity can fix those individually, but as far as I'm aware, no alternative has been shown to be viable once taken in combination.
No one has been able to come up with a compelling alternative model of gravity. All the attempts so far have largely failed (their have been many), as the proposed theories break in one or more critical ways.
I think their may be a small number of languishing models that aren't totally disproven but don't look very promising. But I'm not sure.
Astronomers are scientists, they don't make shit up just out of boredom, they only believe theories when they've withstood rigorous testing through observational evidence.
It's important to understand the history of dark matter / "missing matter". It's a multi-decade history that originally started out with a small amount of intriguing but seemingly persistent observational data that couldn't be explained easily (namely that when you measure how much galaxies weigh by looking at how fast the stars are orbiting that figure differs a lot from the weight you get by calculating up the contribution from all the stars and gas and dust and whatnot that we can see (the "light" matter)). In response a huge number of different "theories" (more properly hypothesis) were brought up to try to explain the mystery, while new ways of studying the problem were thought up as well.
And over the many years after the initial evidence came to light (in the '70s) considerably more evidence came to light from a wide diversity of observations. I won't list them here but I'll point out that the wikipedia page on dark matter has a good run down. Anyway, the fascinating twist to the story here is that as this evidence came to light it started eliminating various hypotheses about this missing matter until finally only one was left: the current theory of dark matter. It's not just a crazy idea, it's a crazy idea that fits all of the evidence when nothing else did.
So don't look at the WIMP dark matter theory as though it's just some half-baked idea, it's a hard fought veteran of numerous campaigns to kill it, but it just keeps going because by all the evidence it seems to be the only theory that explains reality.
As to specifically the "modified gravity" theories that compete with dark matter, they have a hard time explaining several observed phenomena, especially things like the famed "Bullet Cluster". There are a few examples where we can observe collisions between galaxy clusters and through different techniques we can map out the distribution of stars and of gas and of mass in the collision. What we observe in the Bullet Cluster as well as some others is that the gas is in a completely different place than the stars (because stars in galaxies mostly pass through one another whereas gas clouds squish together) and the center of mass of the stars of the galaxies is separated from the center of the non-visible mass of the galaxies. This absolutely cannot be explained by any of the modified gravity theories.
The article describes the rationale, as does the "observational evidence" section of the Wikipedia article on dark matter[0].
tl;dr no alternative models for gravity can sufficiently explain all of the data, but general relativity and "dark matter being particles" does, so that's what we're going with.
