We start with the beliefs we can't doubt, like the laws of logic, tautologies, mathematical truths, the fact of our own conscious experience, and so forth. We then weigh all the possible realities that are consistent with these truths, and go with what's most probable. Most of our beliefs are based on inductive reasoning. If we continue to observe day and night every 24 hours, the cycle will probably continue. Why? How do we justify inductive reasoning?
If the universe was not governed by laws and consistent behaviour, we should observe maximum entropy and pure chaos from the beginning (of course we wouldn't exist in such a universe in the first place). It is possible that out of pure randomness, the universe could behave "as if" it was governed by laws, but the probability of this occurring for billions of years and then stopping is infinitesimally small.
We are constantly proving to ourselves that our beliefs are valid, because they are justified by observation. The theories we use in science allow planes to fly, give us internet access, and take us to the moon. We can predict the positions of the planets years in advance, and the weather days in advance. Aren't we using observation to justify observation? Ultimately yes, but only through theories based upon realism: a belief in a reality independent of observation, to explain the consistency and predictability of observation.
And yes, the universe could be a simulation (I don't believe it is, but this is another question altogether), but our scientific theories would still be true insofar as they provide accurate predictions, and describe the properties of objects in relation to other objects, whatever those objects actually "are". And there would still have to be a mind-independent reality to produce the simulation in the first place. It can't be simulations all the way down.