With Hubble Space Telescope imaging, we investigate the progenitor population and formation mechanisms of the intracluster light (ICL) for 23 galaxy groups and clusters ranging from 3$times10^{13}<$M$_{500,c}$ [M$_odot$]$<9times10^{14}$ at 0.29$<$z$<$0.89. The color gradients of the BCG+ICL become bluer with increasing radius out to 53-100 kpc for all but one system, suggesting that violent relaxation after major mergers with the BCG cannot be the dominant source of ICL. For clusters the BCG+ICL luminosity at r$<$100 kpc (0.08-0.13 r$_{500,c}$) is 1.2-3.5$times 10^{12}$L$_odot$; for the groups, BCG+ICL luminosities within 100 kpc (0.17-0.23 r$_{500,c}$) range between 0.7-1.3$times 10^{12}$ L$_odot$. The BCG+ICL stellar mass in the inner 100 kpc increases with total cluster mass as M$_bigstarpropto$M$_{500,c}$$^{0.37pm0.05}$. This steep slope implies that the BCG+ICL is a higher fraction of the total mass in groups than in clusters. The BCG+ICL luminosities and stellar masses are too large for the ICL stars to come from the dissolution of dwarf galaxies alone, implying instead that the ICL grows from the stripping of more massive galaxies. Using the colors of cluster members from the CLASH sample, we place conservative lower limits on the luminosities of galaxies from which the ICL could originate. We find that at 10 kpc the ICL has a color similar to massive, passive cluster galaxies ($>10^{11.6}$ M$_odot$), while by 100 kpc this colour is equivalent to that of a 10$^{10}$ M$_odot$ galaxy. Additionally, we find 75% of the total BCG+ICL luminosity is consistent in color of galaxies with L$>$0.2 L$_*$ (log(M$_bigstar$[M$_odot$])$>$10.4), assuming conservatively that these galaxies are completely disrupted. We conclude that tidal stripping of massive galaxies is the likely source of the intracluster light from 10-100 kpc (0.008-0.23 r$_{500,c}$) for galaxy groups and clusters.
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