The AGILE satellite detected several episodes of transient gamma-ray emission from Cygnus X-3. Cross-correlating the AGILE light curve with both X-ray and radio monitoring data, we found that the main events of gamma-ray activity were detected while the system was in soft spectral X-ray states, that coincide with local and often sharp minima of the hard X-ray flux, a few days before intense radio outbursts. This repetitive temporal coincidence between the gamma-ray transient emission and spectral state changes of the source turns out to be the spectral signature of high-energy activity from this microquasar. The gamma-ray differential spectrum of Cygnus X-3 (100 MeV - 3 GeV), which was obtained by averaging the data collected by AGILE during the gamma-ray events, is consistent with a power law of photon index {alpha} = 2.0 +/- 0.2. Finally, we examined leptonic and hadronic emission models for the gamma-ray activity and found that both scenarios are valid. In particular, in the leptonic model - based on inverse Compton scatterings of mildly relativistic electrons on soft photons from both the Wolf-Rayet companion star and the accretion disk - the emitting particles may also contribute to the overall hard X-ray spectrum, possibly explaining the hard non-thermal power-law tail seen during special soft X-ray states in Cygnus X-3.