How can agricultural ammonia lead to PM2.5 production in the atmosphere?
Acid/Base Chemistry

Gaseous ammonia (NH$_3$) acts as a base when it dissolves in water. The reactions are below.

NH$_3$(g) + H$_2$O(l) $\rightleftharpoons$ NH$_3$(aq)

NH$_3$(aq) + H$_3$O$^+$(aq) $\rightleftharpoons$ NH$_4^+$(aq) + H$_2$O(l)

The end product is an ammonium ion. We also have gas phase species SO$_2$ and NO$_2$ in air. They dissolve in water to form the conjugates of the acids H$_2$SO$_4$ and HNO$_3$ (e.g. SO$_4^{2-}$).

The protonized ammonium and the deprotonized acids (e.g. NO$_3^-$) are not gaseous anymore. They are also NOT solids. They are dissolved ions.

Precipitation and Nucleation

Ammonium cations react with the acid anions in an acid/base reaction to form a salt complex. An example is ammonium sulfate (NH$_4$)$_2$SO$_4$. When the concentration is low, salts remain as dissolved species in water. As the salt concentration increases, at some point, the salt condenses to a solid (it precipitates out of solution). Salt concentrations increase as the water droplets in the air take on more acid and base species. Salt concentrations also increase as the water droplets evaporate. This causes the water droplets to act as sites for the formation of solid salt particles.

Air also contains other solids as ultra-fine particles. Think ... dust.  

All solid surfaces in water act as sites to catalyze the further capture of dissolved salts. They act as proton exchange mediators, better than the liquid phase itself.

The process of formation of a solid particle or capture of a dissolved species onto a solid surface is called nucleation. Nucleation is followed by growth. Some of the solid salts can alternatively redissolve when the conditions are reversed ... e.g. the concentration of the salt in solution surrounding it decreases.

The figure below illustrates the entire process using ammonia and nitric acids (formed from nitrogen dioxide).



Image caption: The compounds in green colored font are located at (mostly wet) particle surfaces or are dissolved in droplet water. Compounds written in black colored font are gaseous. A second arrow head is missing at the arrow connecting gaseous $HNO_3$ and particle-bounds $HNO_3$. Gaseous $NH_3$ is missing in this figure.

Because proton-exchange favorably takes place at surfaces or in droplets, the particles grow favorably rather than redissolve (see below).

References

An exemplary particle size distribution is shown below (own work (Neumann, 2016, Fig. 2.1) but content based on Seinfeld and Pandis (2006a, Fig. 2.7,p.59) which is based on Whitby and Cantrell (1976)). The Y-axis shows particle abundance (roughly; please see Fig 2.2 in the linked document above for details).



The major source of the coarse particles on the right are primary emissions of particles (dust, sea salt, ...). The accumulation mode particles in the center are partly caused by primary emissions (fine sea salt, combustion processes, ...) and partly grown from smaller particles (Aitken mode or ultra-fine particles). Actually, the ultra-fine particles are grown from so called nucleation mode particles by the condensation of further atmospheric gaseous compounds. The nucleation mode is missing here (and merged with the Aitken mode). Nucleation mode particles are those particles, which are formed by ammonia and atmospheric acids.

Other sources for these nucleation mode particles are gaseous organic compounds, which lump together to form particles. The latter particles are also called SOA (secondary organic aerosol). Isoprene is one of the compounds that forms SOA.

Health Impacts

When we combine air polluted by combustion emissions ($NO_X$) with air affected by agricultural emissions, we can expect the formation of ammonium nitrate particles ($NH_4^+$ + $NO_3^-$). Ammonium nitrate can irritate the eyes, nose and lungs. As stated in a comment and to the best of my knowledge, ammonium nitrate particles are not as harmful as e.g. fine soot particles. But I am not a expert on this field.

Ammonium sulfate (from $SO_2$ from combustion emissions) have been shown to have negative health impacts on the respiratory system but also on the cardiovascular system (heart etc.; if the particles are sufficiently small to pierce the lung-blood barrier).

Ultra fine ammonium nitrate particles can be nuclei for condensation of further gaseous compounds -- e.g. VOC (volatile organic compounds) --, which might have harmful impacts on our respiratory system.

notes

Commonly, atmospheric particles are not dry spheres/lumps of matter but somehow associated with water: water droplets with dissolved compounds or a solid core with water attached to the surface; whereas dissolved compounds might be dissolved in the water of the latter type.

The atmospheric life time of gaseous acids is very short because they tend to get attached to water (and deprotonize -- reducing the pH value of the water). Partly, the reaction from precursors to acids takes place in the wet particle/droplet phase. E.g. most of produced sulfuric acid ($H_2SO_4$) takes place in the wet phase.

tl;dr

Thus, we foster the growth of ultra-fine nuclei by the condensation of ammonia (as base) and condensation/formation of acids. 

References:


Seinfeld and Pandis, 2006: Atmospheric Chemistry and Physics: From Air Pollution to Climate Change.
Whitby and Cantrell, 1976: Fine particles in Proc. Int. Conf. Environ. Sens. and Asses.