Some of it is just atmospheric and background storytelling. The shopowner's skeleton in a bathtub with a toaster and a fork tells of the despair that some felt. There are so many with obvious suicide implements (guns, knives, pills, etc) near them when you find the skeletons that it tells a story in and of itself.
The primary strategic nuclear device that was fired at the Commonwealth 'missed' the city and landed in what is now the Glowing Sea (and may have instead been targeting the Sentinel Site, but was too late as that site fired its missiles successfully). That means that while many did die from the initial explosion, heat blast, and then shockwave the vast majority of Boston lived through it.
It's the radiation and the collapse of society that killed them. Go to a coast line in south Boston and along downtown and all the residential areas. There are wheelchairs of the sick who were brought to the shore to watch the sea as they died. The hospitals are full of them... people waiting for help that never came. Churches full of corpses as they prayed for salvation.
Then there's the madscrabble areas. Train cars that were once triage stations, now overrun with ghouls, but with a table full of teddy bears and a sign that promises that all are welcome and safe there. They're everywhere, all over the map.
Boston died a slow, miserable death. Sick and dealt a mortal blow, it shuddered and wept as it died.
Oh... and the only side quest that I've ever seen in a game that made me cry (every time I've done it on three characters) is the non-quest 'quest' of finding the Holotape of Cheryl and Marlene Glass. If you go to the Slog and talk to Arlen Glass, you have the option of giving it to him.
At that point is some of the most soul crushing and tragic dialog I've seen in Fallout. In a good way. It really hits you hard in the gut what sort of loss people have experienced. That he just sits there afterwards, listening to it over and over again, weeping, is all it takes to tell you everything. He stays like that until you leave and return.
Well done, Bethesda.