How do we know the current state of the universe by observing it from Earth?

Since light travels at a constant and finite speed, looking into the Universe means looking back in time.

No matter in which direction we look, on average we see the same: Locally, we see evolved galaxies, far away we see young galaxies, and even farther we see the so-called cosmic microwave background, which is light from so far back in time that galaxies hadn't yet formed.

That is, the Universe seems to be isotropic. Unless we occupy a special place in the Universe, this implies that the Universe is also homogeneous, i.e. it would look more or less the same if you were located in another part of it.

This is turn means that the current state of the Universe can be inferred by observering the local region. The most distant observed galaxies can then be concluded be in the same state (statistically speaking; of course they don't look exactly the same, but any given property, e.g. the distribution of their sizes, will be the same as in our neighborhood).