We analyze the CIII]-λ1908 Å emission properties in a sample of 3899 star-forming galaxies (SFGs) at 2 < z < 3.8 drawn from the VIMOS Ultra-Deep Survey (VUDS). We find a median rest-frame equivalent width EW(CIII]) = 2.0 ± 0.2 to 2.2 ± 0.2 Å for the whole SFG population at 2 < z < 3 and 3 < z < 4, respectively. About 24% of SFGs are showing EW(CIII]) > 3 Å, including ∼20% with modest emission 3 < EW(CIII]) < 10 Å and ∼4% with strong emission EW(CIII])> 10 Å. A small but significant fraction of 1.2% of SFGs presents strong CIII] emission 20 < EW(CIII]) < 40 Å; the four strongest emitters (EW(CIII]) > 40 Å up to ∼95 Å) are associated with broad-line AGN. While this makes CIII] the second most frequent emission line in the UV rest-frame spectra of SFGs after Lyman-α, this line alone cannot be considered an efficient substitute for measuring a galaxy redshift in the absence of Lyα emission, unless the spectral resolution is R > 3000 to distinguish among different possible doublets. We find a large dispersion in the weak correlation between EW(CIII]) and EW(Lyα), with galaxies showing strong CIII] and no Lyα, and vice versa. The spectra of SFGs with 10 < EW(CIII]) < 40 Å present strong emission lines that include CIV-λ1549, HeII-λ1640, and OIII-λ1664, but also weaker emission features of highly ionized elements such as SiIV-λ1403, NIV-λ1485, NIII-λ1750, or SiIII-λ1888, indicating the presence of 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 increasingly powerful outflows with velocities up to 1014 km s⁻¹; this is beyond what stellar winds are commonly producing. The strongest CIII] emitters are preferentially located below the main sequence of star-forming galaxies; the median star formation rate is reduced by a factor of two. In addition, we find that the median stellar age of the strongest emitters is ∼0.8 Gyr, which is about three times that of galaxies with EW(CIII]) < 10 Å. X-ray stacked imaging of the strong CIII] emitters sample show a marginal 2σ detection that is consistent with low-luminosity AGN log(L_X(2−10 keV)) ∼ 42.9 erg s⁻¹. Previously presented spectral line analysis and classification support that the strongest emitters require the presence of an AGN. We conclude that this complementary set of evidence is indicative of significant AGN feedback acting in SFGs at 2 < z < 3.8, and it strongly contributes to star formation quenching. We find that quenching timescales of ∼0.25−0.5 × 10⁹ years are necessary for this AGN feedback to turn part of the star-forming galaxy population with M_star > 10¹⁰ M_⊙ at z ∼ 3 into the population of quiescent galaxies observed at redshift z ∼ 1−2.