big bang theory - Can it be inferred that our cosmological horizon has increased over time?

There are a couple of different horizons you should care about. The first is the cosmological horizon, which is the furthest you could possibly see given that photons travel at a finite speed and the universe is not infinitely old. Since nothing travels faster than the speed of light, this is quite literally the furthest we could ever hope to see - everything outside is causally disconnected from us (though this horizon does increase with time, it asymptotically reaches some ultimate distance due to the accelerated expansion of space-time).

Practically speaking, the horizon which we really care about is the cosmic microwave background. This is the point at which the universe became cool enough such the atoms could remain neutral. Why is this the important horizon? Before this, photons simply could not travel very far before interacting with charged particles. Before the surface of last scattering the universe was essentially opaque (it would be like looking through a cloud). Afterwards, however, it was transparent. Everything from the surface of last scattering and later (redshift of $zsim 1100$ and lower) is said to be part of the observable universe.

Though the universe may still be infinite in size, we will never be able to see these objects (if they exist) if they are passed this horizon. The other thing to remember is that the further you look, the further back in time you're seeing. Galaxies did not always exist. It took time to form larger overdensities like stars, galaxies, and clusters from small perturbations in density during the early universe. After the surface of last scatter, there was a period of time in which the universe was very dark and potentially very boring from an astronomer's point of view. This is the time after atoms became largely neutral yet before the first stars and galaxies "turned on".