We present new measurements of Cosmic Infrared Background (CIB) anisotropies using Planck. Combining HFI data with IRAS, the angular auto- and cross-frequency power spectrum is measured from 143 to 3000 GHz. After careful removal of the contaminants (cosmic microwave background anisotropies, Galactic dust and Sunyaev-Zeldovich emission), and a complete study of systematics, the CIB power spectrum is measured with unprecedented signal to noise ratio from angular multipoles ℓ ~ 150 to 2500. The interpretation based on the halo model is able to associate star-forming galaxies with dark matter halos and their subhalos, using a parametrized relation between the dust-processed infrared luminosity and (sub-)halo mass, and it allows to simultaneously fit all auto- and cross- power spectra very well. We find that the star formation history is well constrained up to redshifts around 2, and agrees with recent estimates of the obscured star-formation density using Spitzer and Herschel. However, at higher redshift, the accuracy of the star formation history measurement is strongly degraded by the uncertainty in the spectral energy distribution of CIB galaxies. We also find that the mean halo mass which is most efficient at hosting star formation is log(Meff/M⊙) = 12.6 and that CIB galaxies have warmer temperatures as redshift increases.