Hi Kelly. 

A woman talked to a person in such a way about her own humility and piety, that the person
would  refuse the opinion of anyone who stubbornly disbelieved that she was actually humble and pious.

"Often and often has she talked to me and the neighbours regarding her own humility and piety",

She talked to him and the neighbours about her humility and piety

"pointing them out in such a way"

She was pointing them out (the humility and piety) in such a way

"that I would defy the most obstinate to disbelieve her"

that the person here "the narrator" would defy all the stubborn people that did not believe her.

I hope my explanation help you :).

The Luck of Barry Lyndon
By William Makepeace Thackeray