Kate Hardcastle knows that the wealthy Sir Charles Marlow is shy and tongue-tied around women of his own class, but a completely different person, free and easy, around lower-class women. In order to encourage him to woo her, she pretends to be a barmaid. If he believes Kate is a barmaid, he will feel comfortable with her and able to be his more confident self. She "stoops" or pretends to assume a lower position on the social scale, in order to "conquer" or win love and marriage.
I am not sure what "critically justify" means, but the title is clever and apt, encapsulating the key concepts of the play. It's an elegant eighteenth century title, with a sense of paradox created in the balancing and juxtaposing of seemingly opposite images: stooping and conquering. It leads one to question how a person could conquer through stooping, and as such, reveals that power manifests in different forms. The title could also be "critically justified" as highlighting how often we react to people in terms of their outward status rather than their inward selves. Kate is Kate whether a barmaid or a lady, but Marlow, at least initially, can only respond to the outer wrapping. The play is still remarkably relevant and funny today, showing that we yet allow status to intimidate us.
The title of this novel refers to the "stooping down" of Kate Hardcastle from her position in high society to the position as a barmaid. She does this in order to test the feelings of Marlow, to make sure that he loves her for herself and not for her money. In the end, she gets what she wants, and proves a point. She learns that Marlow's feelings are genuine and demonstrates that love is not controlled by social position. By "stooping down", she conquered society.