Multiwavelength deep observations are a key tool to understand the origin of the diffuse light in clusters of galaxies: the intracluster light (ICL). For this reason, we take advantage of the Hubble Frontier Fields (HFF) survey to investigate the properties of the stellar populations of the ICL of its six massive intermediate redshift (0.3 < z < 0.6) clusters. We carry on this analysis down to a radial distance of ˜120 kpc from the brightest cluster galaxy. We found that the average metallicity of the ICL is [Fe/H]ICL ˜ -0.5, compatible with the value of the outskirts of the Milky Way. The mean stellar ages of the ICL are between 2 and 6 Gyr younger than the most massive galaxies of the clusters. Those results suggest that the ICL of these massive (>1015 M⊙) clusters is formed by the stripping of MW-like objects that have been accreted at z < 1, in agreement with current simulations. We do not find any significant increase in the fraction of light of the ICL with cosmic time, although the redshift range explored is narrow to derive any strong conclusion. When exploring the slope of the stellar mass density profile, we found that the ICL of the HFF clusters follows the shape of their underlying dark matter haloes, in agreement with the idea that the ICL is the result of the stripping of galaxies at recent times.