Drywall looks solid and dense, why can I hear voices from the other side of the wall?
Drywall is nothing but gypsum covered with paper, essentially chalk and paper. Paper and gypsum are great conductors of vibrations and sound. Unfortunately drywall doesn't have a sound stopping power. The STC value of drywall is actually low.
To understand which building materials might help with sound travelling from room to room, refer to this page on soundproofing for walls.
Isn't there insulation in the wall?
If the wall is an interior wall, insulation isn't required, by code. If the studs are tin, which most are in interior office walls, the tin will actually act as an amplifier, much like playing telephone with two tin cans and string.
If there is insulation in the wall, there insulation will only serve to dampen the sound, the sound will still travel through it.
What can be done to help stop the transfer of sound from one room to another, through the drywall?
Basically, the drywall would have to come down on one or both sides, leaving the studs exposed. A contractor would then install mass loaded vinyl, which is a dense synthetic. The Contractor should opt for 5/8" drywall, and then they would surround the receptacles with acoustic putty, to prevent sound leakage.
To gain a better understanding of the acoustic science of soundproofing read this page.
If you are just concerned with stopping sound coming in through one wall, read more have about soundproofing walls.
How about the sound under the walls?
To be thorough, you will want to have the wall, decoupled from the ceiling and the floor. As odd as that sounds, it will stop the sound vibrations travelling from the wall into the floor and ceiling.
Is there anything that can be applied to the outside of the drywall instead of getting involved with demolition and construction?
Unfortunately the product installed directly on the surface of the drywall would have to be significantly thick, over 1/4" thick to have an effect. That product would also need a air gap between it and the drywall.
Is there a cheap quick solution?
There is no cheap and easy solution, I would advise that you check that sound isn't bleeding into the room from other sources first, such as underneath the door, or through the HVAC.