Use of western blotting
They are doing a peptide competition assay (see abcam protocol).

The antibody is incubated with a purified sample of the antigen and used in a normal Western-Blot of cell lysates. Any cross-reactivities of the antibody would still be visible as only the interaction with the epitope of interest is blocked. So all bands that disappear after such treatment are indicators for the substance that was used for blocking. A Western-Blot with non-blocked antibody is normally used for comparison.

In the experiment you mentioned, they vary the principle a bit, as they are comparing two blots - one blocked with a succinylated library, one with a non-succinylated one. As the band disappears after treatment with the succinylated library, they confirm that their antibody is specific against succinylated proteins - at least in these three cases.