We already know that the electric fields in a thunderstorm can exceed 1-1.5KVolt/cm--that's why you get lightning. One electron knocked loose is accelerated in that field and knocks loose others, and others, and pretty soon you have a nice flash in the sky.
The cosmic rays we see at sea level are mostly just muons left over from more extensive showers of particles high in the atmosphere that came about when some cosmic proton or helium nucleus (or oxygen or iron or) hit a nucleus in our atmosphere. A charged particle flying though the air at high speeds is going to ionize some molecules along the way--providing the electrons needed to start a local cascade.
The experimenters looked for "Extensive Atmospheric Showers:" muons reaching the earth over a wide area and Cerencov radiation from the high altitude particles flying through the upper atmosphere but not reaching the ground. And they looked at lightning flashes, and found some coincidences. Cosmic ray showers can trigger lightning.
They also estimated the energy of the original cosmic ray for an "EAS", and looked at the energy of the radio pulse from it, and estimate that the radio pulse resulting from a cosmic ray shower going through a thunderstorm is many times larger--the thunderstorm amplified it.
FWIW, other work has suggested that the ionization trails from cosmic rays can form "seeds" for water droplets to begin forming, which could have an effect on cloud formation. There is no good reason why cosmic ray rates should be constant over the long term, and some good reasons why they should change--nearby supernovae, for example, could kick the rate way up--and presumably cloud formation rate as well. The atmosphere might be more reflective and the Earth correspondingly cooler.
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