The word literally means "new building" (building as a gerund, not as a noun). The word was coined when they started building appartments in blocs, up to four floors high (that's five American).
Why four? The standards demanded an elevator if higher.
There was a world of difference between living in a house and in novogradnja. A house has a garden, yard, lawn, shed, attic, maybe a cellar. An apartment in novogradnja has only the rooms, a few square meters of balcony, and maybe as much space in the cellar (which anyone can break into). Your territory ends at your door - beyond that, there's the staircase with the lights on a short timer, so if you're on the 3rd floor, you have to hit at least two switches on your way down, or else you'd find yourself in the dark. Garbage chutes weren't standard until the seventies, so you'd have to carry it down. Central heating also wasn't an option in those earlier buildings, so you'd have to carry coal upstairs from the cellar. Your mailbox was downstairs and was often ransacked by the kids. Also, anything larger than a regular envelope would be forcibly crammed into your mailbox, because the postman wouldn't walk up - if it was an LP record, too bad.
Most important, you didn't own it - you had a legal right of residence after a while (5 years, IIRC) and then they couldn't kick you out even if you stopped paying the rent, which was symbolic anyway. Since you didn't own it, you weren't in charge of maintenance either, so whatever broke (and it did), you'd have to wait for the housing enterprise or whoever was in charge to send a majstor.
Yet everyone wanted to live in those, because it was basically free. Your rent wasn't higher than your power bill, you had no worry about the maintenance, cleaning outside your rooms. And initially the walls were all solid, so you wouldn't hear much of your neighbors (in the seventies, however, the beauties of sheetrock over a frame were discovered).
In the nineties these apartments were sold to the tenants at ridiculous prices. For an amount ranging between a month's and six month's worth of gasoline for your car you could buy off your place... and enter into something like a condo relationship with your neighbors. And you were still in novogradnja and not expected to shovel the snow from your pavement (and get fined by the city if you don't) like those guys in the houses across the street.