We were introduced to Storm Troopers serving the emperor and Darth Vader in the early Star Wars films. But the Empire fell and now we see Storm Troopers serving Leader Snoke and Kylo Ren in The Force Awakens. What back story is given to explain why the same army with the same body armor survives to serve two entirely different governments?
Just because they wear similar gear does not make them the same army.
In the film, Finn, Hux, and Phasma all emphasize that these particular stormtroopers were conditioned from birth to be completely obedient to the First Order.
This suggests that the First Order was established first, and then its army was created. They happen to be using a similar (but not identical) uniform to the Imperial stormtrooper regiments, at least partly for ideological reasons. The point is that these new stormtroopers were trained for the First Order. There is no evidence that previous Imperial stormtroopers had become First Order stormtroopers.
After the death of Vader and the Emperor, the Empire didn't just disappear in a puff of smoke. Just as with any government, others swooped in to take power where they could, and the Empire lingered on and continued fighting, even though they had suffered a crippling blow. Think of the First Order not so much as a different government, but rather an evolution of the old.
It's also not unusual for a new government to use the iconography of the past to project a certain image. The First Order obviously wanted to have that connection to the Empire in people's minds, so naturally, they kept the look for that purpose as well.