With each rank of this perk, your character will do 20% 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 18.104.22.168xx, 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.
“The bigger the boom, the better! At the highest ranks, a proficient expert can craft a potent bomb or ten and lob them all at even the most dangerous of marauding foes, dropping the enemy before engaging in closer combat. Are you a fan of mines? Do you enjoy using V.A.T.S. to shoot grenades or mines close to or carried on an enemy? Then think about this perk.”— Fallout 4 Vault Dweller's Survival Guide description
Demolition Expert increases the damage dealt by thrown grenades, mines, explosive weapons such as the missile launcher and Fat Man (these also stack with the Heavy Gunner Perk) and even the legendary Explosive prefix.
However, it does not increase the secondary damage dealt by said weapons (e.g. a grenade that deals 2 types of damage). In this case, it will only increase the damage of the first type. That also means that this perk does not increase the damage of pulse mines and pulse grenades because they only deal secondary energy damage.
The "gain a throwing arc" for rank 2 actually means that when your player character primes a grenade, a giant green arrow will show the trajectory and initial landing point for that grenade. It does not account for successive bounces, but even still can be a great aid in successfully using grenades, especially since grenades cannot be used in VATS anymore.
Since it's not completely clear from the rank descriptions, ranks higher than 1 are still needed to unlock successively more advanced crafting recipes.
Rank 3 adds 25% explosion radius.
This perk does NOT affect the "Explosive Vent" power armor mods.
It does, however, affect the "Explosive" legendary weapon effect to great effect. See the respective page for details.
This perk also unlocks the ability to craft certain explosives at workbenches.
Prior to Fallout 76 patch 22.214.171.124, the damage bonuses provided by each rank of the Demolition Expert perk were +20%/40%/60%/80%/100%, but afterwards were reduced to +20%/30%/40%/50%/60% to match other weapon perk damage bonuses.