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 220.127.116.11xx, 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.
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 (for e.g. a grenade that deals 2 types of damage). In this case, it will only increase the damage of the first type. In addition, this perk does not increase the damage of pulse mines and pulse grenades for some reason.
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.
Rank 3 is currently bugged. Taking rank 4 will shut off the increased range that it adds. For PC users, a fix is available here