  We analyze the CIII]-1908\AA 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$0.2\AA ~for the SFG
population at 2<z<3 and 3<z<4, resp. About 24% of SFGs are showing
EW(CIII])>3\AA, including 20% with 3<EW(CIII])<10\AA ~and 4% with strong
emission EW(CIII])>10\AA. A significant fraction of 1.2% of SFGs presents
strong CIII] emission 20<EW(CIII])<40\AA. This makes CIII] the second
most-frequent emission line in the UV-rest spectra of SFGs after
Lyman-$\alpha$. We find a large dispersion in the weak EW(CIII]) - EW(Ly\alpha)
correlation, with galaxies showing strong CIII] and no Ly$\alpha$, and
vice-versa. SFGs with 10<EW(CIII])<0\AA ~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])<10\AA. 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.
