Also, notice how Sheila and Eric are affected by it and start to change their views? That wasn't change, it was part of their conscience all along, but it took a direct confrontation with their consciences in the form of Inspector Goole to realise it. Mr Birling, Mrs Birling and Gerald, on the other hand genuinely believe that they had nothing to do with it and weren't responsible, and therefore were not affected.
WMG / An Inspector Calls
The Inspector is a ghost, or a recreation of the Birling's (and Gerald's) consciences.Think about it — The Reveal raises the point that no-one has actually seen this "Goole" person, which raises the suspicion as to if the Inspector is actually real or not. Also, Sheila mentions that the Inspector seemed to know everything before the Birlings even revealed their secrets. How could he have possibly known about Gerald's affair and Eric's drunken night out during which he impregnated 'Eva'? Simple: If he was part of their consciences, then it would make perfect sense for him to know.
They're all dead.The Birling's actually all died of food poisoning at dinner that night, and this is their personalized Hell.