With each rank of this perk, your character will do 20% percent more damage with all weapons governed by the Explosives skill.
Note that this damage bonus only affects the actual explosive part of the weapon's damage, not the physical impact. For grenades and mines this is a negligible difference as they typically only deal 5 physical throwing damage at most. But for a Fat Man (GRA) it is much more significant as 400 of its 1000 base damage is from the physical impact.
Prior to patch 188.8.131.52xx, the Demolitions Expert perk only applied to weapons listed in a specific Form List, which contained a few notable (and undesired) omissions. The patch subsequently expanded the Demolitions Expert perk to apply to all weapons governed by the Explosives skill, unique, DLC, or otherwise. For reference, a full listing of Explosives-skill weapons follows.