This is a slightly more technical answer that should nonetheless illustrate the concepts.
Wind back the clock to 1845. You are an astronomer of that period, and you are familiar with Newton's Laws, which have been extensively verified to work in the lab. You observe Uranus and notice that it moves largely as Newton's Laws predict it to, but there is some disparity with the theoretical prediction. What could be causing the discrepancy? The two obvious explanations are:*
- Newton's Laws aren't actually right. They work in the lab, but once you extend to 2.9 billion kilometers (the average distance from the Uranus to the Sun), it no longer works.
- There's something else "out there" that is exerting extra gravity and causing the discrepancy.
With option 2, you can go further: if you assume that Newton's Laws are right, then you can calculate more properties of the extra something: its mass, its orbit, and so on. But note Newton's Laws only say the how much mass the new body has. It doesn't say what the new body is. It could be a new planet, or it could be something else. This is how Neptune was discovered - it was a theoretical prediction based on Newton's Laws.
The situation with dark matter is similar. If you assume General Relativity is correct, then there's a discrepancy between theory and observations that can be resolved by postulating the existence of dark matter. You don't know what dark matter is, but you can calculate how much dark matter there should be (which is where the 85% came from).
Or you could say General Relativity is incorrect, which might also invalidate the need for dark matter. This also happened historically - Mercury shows precession that Newton's laws could not explain, leading some to conjecture that there must be another planet closer to the Sun than Mercury. The missing planet (Vulcan) was never found, and General Relativity removed the need for the planet. In the same way, if MOND turns out to be correct, then dark matter would be unnecessary, and the statement that dark matter makes up 85% of the matter content would be incorrect.
At present, most astrophysicists believe dark matter is strongly favored over alternative explanations, which is why they are happy to make the statements you allude to. The preponderance of evidence is in favor of dark matter** - but they might all get egg on their faces if alternative gravity turns out to be right.
*Of course, in practice, it's significantly more complicated. Confounding factors could be: perhaps Uranus's mass was estimated incorrectly, or the distance to Uranus was estimated incorrectly, or your observations of Uranus are incorrect because of some atmospheric fluctuation in the Earth, etc.
**I suspect what really convinces most astrophysicists is the sheer number of independent lines of evidence that can all be explained by standard cold dark matter. An alternative hypothesis might be able to explain any individual line of evidence, but explaining all of them with one framework is extremely difficult. No alternative theory has come close, but as long as dark matter is not identified, some people will continue to think about them.