The radio room does two things: raise happiness for other dwellers and occasionally attract outsiders to join the line outside your vault.
I have not tested the happiness mechanics in any depth, but that functionality is described on a loading screen so I assume it works in game.
Attracting outsiders works a bit differently from all the other resource production rooms. The standard resource rooms run for a set time, then flash with an icon when the timer is done and you (or Mr. Handy) can tap the room to collect the resources. Merging the same room into a bigger room increases the time it takes to generate the resources, but also increases the total amount of resources collected at the end. Staffing the room with dwellers high in the room's stat decreases the time for the resource cycle to complete.
The radio room is different. Each time it completes its cycle, it has a chance to attact a single new dweller. If it fails that chance, it immediately starts a new cycle, whether you have the vault open or not. It repeats this cycle until it successfully attacts a dweller. When that happens, it pops the icon and waits for you to tap it. Merging radio rooms together and upgrading the room both increase the chance of a new dweller at the end of the cycle as well as increase the time each cycle takes. Staffing the room with high charisma dwellers decreases the time taken per cycle, but does not affect the percentage chance per cycle.
What this boils down to is that a fully upgraded, full size radio room with six dwellers of 10 charisma each should attract a new dweller faster than a small radio room with a single low charisma dweller, both because each cycle is faster and because each cycle has a higher chance of attracting an outsider.
As a point of reference, my vault with a top of the line radio room with 6 dwellers of 14 charisma each (10 stat + 4 cha outfit) consistently attracts a new dweller every 6-8 hours or so. I send the outsider to die in the wasteland if their stats are not what I'm looking for, but that's a different thing. It's also rumored that assigning Three Dog to the radio room does something special, but I haven't confirmed that.
I also have not tested multiple radio rooms in the same vault.
Yeah, it's that slow.