What does these $\Delta \Delta G$ numbers signify?
Very basically, what they've done in the paper was incubate different DNA sequences with histone octamers which were then separated on a gel. From this they could measure the ratio between nucleosomal and free DNA, which represents an equilibrium constant. This can then be expressed as a change in free energy ($\Delta G=-RTln(K)$), which is what the table shows. A lower (more negative) free energy change corresponds to a higher equilibrium constant which represents a greater proportion of DNA bound to histones. In other words, the free energy change quantifies the propensity of these different DNA sequences to form nucleosomes.