Context, or AO3, matters more in An Inspector Calls than in almost any other set text, because the play is really a message from Priestley to his own time. The single most useful thing to understand is the gap between when it is set, 1912, and when it was written and first performed, 1945. Everything Priestley does with that gap is designed to make a point.
1912: the confident world before the wars
The play is set in 1912, in a rigid, class-divided society that felt stable and permanent to the wealthy people at the top of it. Priestley uses Mr Birling to capture that misplaced confidence, and to destroy it. Birling calls the Titanic 'unsinkable' and insists there will be no war, both of which the audience knows to be catastrophically wrong. This dramatic irony makes Birling look foolish, so that when he preaches selfish capitalism, we already distrust everything he says.
1945: the audience Priestley was really writing for
Priestley wrote the play in 1945, at the very end of the Second World War, when Britain was deciding what kind of society to build next. The country had just lived through two devastating wars and mass unemployment, and in 1945 it voted overwhelmingly for change and the beginnings of the welfare state. The Inspector's warning about 'fire and blood and anguish' would land on an audience who had just experienced exactly that. Priestley is urging them not to slip back into the selfish attitudes of 1912.
Priestley's socialism
J.B. Priestley was a committed socialist who believed strongly in social responsibility, the idea that everyone has a duty to look after everyone else, especially the poor and vulnerable. The Inspector is essentially Priestley's own voice, arguing this case against Mr Birling's capitalism. Understanding that the play is a piece of political persuasion, not just a family drama, is central to a strong answer.
Class and gender in 1912
Edwardian society was sharply divided by class and gender. The wealthy had almost total power over the working class, who had few rights and little protection, which is why the Birlings can use and discard Eva so easily. Women, meanwhile, were expected to be decorative and obedient, and working-class women like Eva were especially vulnerable to exploitation. Priestley uses Eva's fate to expose the injustice built into this system.
See how this context deepens each theme on the themes page, especially responsibility, and connect it to the Inspector and Mr Birling pages.