There is observational evidence for space expansion, called redshift: Most galaxies are moving away from us and we can measure that by the amount they are redshifted. There are more details here. But anyway, this is exactly what the author seems to deny as evidence.
Hubble noted that distant galaxies had higher redshifts and fitted his observational data with the famous linear relation now known as Hubble's law:
where z is the redshift, c is the speed of light, r is the distance and $H_0$ is Hubble's constant. The further away a galaxy is, the faster is receding from us.
The question is, why are galaxies receding from us according to Hubble's law? How do we interpret this equation?
There used to be the "tired light hypothesis", which tried to explain the Hubble relation by assuming that the galaxies are not receding from us, but that photons simply lose energy as they move through space. This idea quickly fell out of fashion.
Another hypothesis is that the universe is expanding, which is exactly what we count as a fact nowadays.
The key point is that galaxies are not just receding from us, but also receding from each other. We know that from the Cosmological Principle, which states that the universe is homogeneous and isotropic. We do not live in a special place in the universe, but instead the universe is pretty much the same everywhere. There are tons of evidence supporting the Cosmological Principle, and most of established cosmology is based upon its validity.
Philosophically speaking, we do not know that the universe is expanding, as we also do not know that the Cosmological Principle is valid. If you analyze any scientific evidence philosophically, nothing would count as evidence. Yes, galaxies could be just receding from each other due to some random cause, or maybe due to God, but the scientific evidence is clear: The universe is expanding and this is supported by both General Relativity (the Friedmann equations) and observational data (Redshift). What we see is exactly what you would expect from a universe that is undergoing a homogeneous and isotropic expansion. If you reject that the universe is expanding, or that the cosmological principle is valid, you may as well reject all of Cosmology.