This is the Royal Society's press release on the publication of Svensmark et al.:
“Using a box of air in a Copenhagen lab, physicists trace the growth of clusters of molecules of the kind that build cloud condensation nuclei. These are specks of sulphuric acid on which cloud droplets form. High-energy particles driven through the laboratory ceiling by exploded stars far away in the Galaxy - the cosmic rays - liberate electrons in the air, which help the molecular clusters to form much faster than atmospheric scientists have predicted. That may explain the link proposed by members of the Danish team, between cosmic rays, cloudiness and climate change.”
You can also find more about it in the Danish National Space Center press release
The experiment itself is a 7m3 cloud chamber containing marine like air. By controlling the ionization rate using different UV fluxes (i.e., mimicking different cosmic ray ionizations), the group checked to see whether the growth rate of small condensation nuclei is affected by different ionization rates, and indeed it does as can be seen in the first figure. The experiment itself can be seen in the second figure below.
The SKY experiment unequivocally demonstrates that more cosmic rays would imply more condensation nuclei in the atmosphere. In many circumstances, it is reasonable to expect that more of the small condensation nuclei would also imply more of the larger cloud condensation nuclei which form the seeds of clouds. Namely, there would be a link between atmospheric ionization (i.e., cosmic rays flux variations) and climate, in regions devoid of other condensation seeds (such as dust over land).
This link implies that a large fraction of the 20th century global warming can be explained through increased solar activity (which reduced the cosmic ray flux reaching Earth). More about that can be found in this discussion, or in this paper.
Kudos for Henrik Svensmark and his group!