He says that as "space expands the number of particles contained within a volume of space bounded by the horizon is therefore going up year by year as the horizon expands to encompass more and more matter." I thought that number of 10^80 particles was fixed and it neither increases or decreases.

First, keep in mind Davies is speaking of the comoving patch - the observable universe. Because our spacetime is finite in age, and light is finite in speed, we have a particle horizon, the point at which we could see no further ('see' may not be a good term, as light from this region is extremely redshifted.). This point is where we see the cosmic microwave background, the first light emitted.

So, as time goes by, the observable universe gets larger - more light can reach us, and we can observe regions we could not see before. Also, the observable universe grows larger because of expansion. That's why it has a radius of 46.5 billion light years, compared to the 13.7 billion light years you may expect. 10^{80} is the estimated number of nuclei in the observable universe - that is all we can ever speak of. This increases through time, as our observable portion makes up a larger piece of the universe, so we can observe more particles.

If you consider the universe as a whole, atoms weren't even around at the time of the big bang. Electrons weren't held in orbit by nuclei until ~380,000 years after the bang.

The number of atomic nuclei has decreased over the history of the universe - 1] 'naked' neutrons only 'live' for about 880 seconds. 2] fusion combines nuclei resulting in fewer total nuclei.