We analyze the CIII]-1908AA emission properties in a sample of 3899 star-forming galaxies (SFGs) at 2<z<3.8 from the VIMOS Ultra-Deep Survey (VUDS). We find a median EW(CIII])=2.0$pm$0.2 to 2.2$pm$0.2AA ~for the SFG population at 2<z<3 and 3<z<4, resp. About 24% of SFGs are showing EW(CIII])>3AA, including 20% with 3<EW(CIII])<10AA ~and 4% with strong emission EW(CIII])>10AA. A significant fraction of 1.2% of SFGs presents strong CIII] emission 20<EW(CIII])<40AA. This makes CIII] the second most-frequent emission line in the UV-rest spectra of SFGs after Lyman-$alpha$. We find a large dispersion in the weak EW(CIII]) - EW(Lyalpha) correlation, with galaxies showing strong CIII] and no Ly$alpha$, and vice-versa. SFGs with 10<EW(CIII])<0AA ~present strong emission lines including CIV-1549, HeII-1640, OIII-1664, but also weaker emission of highly ionized elements SiIV-1403, NIV-1485, NIII-1750, or SiIII-1888, indicating a hard radiation field. We present a broad range of observational evidence supporting the presence of AGN in the strong CIII] emitting population. As EW(CIII]) is rising, we identify powerful outflows with velocities up to 1014 km/s. The strongest CIII] emitters are preferentially located below the SFGs main sequence, with the SFR reduced by x2. In addition, the median stellar age of the strongest emitters is ~0.8 Gyr, three times that of galaxies with EW(CIII])<10AA. Spectral line analysis presented in a joint paper by Nakajima et al. (2017) confirms that the strongest emitters require an AGN. We conclude that these properties are indicative of AGN feedback in SFGs at 2<z<3.8, contributing to star-formation quenching. We find that quenching timescales of 0.25-0.5x10^9 years are necessary for such AGN feedback to turn part of the star-forming galaxy population with Mstar>10^10 MSun at z~3 into the population of quiescent galaxies observed at redshift z~1-2.
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