So in episode 19 of the anime Nezuko is told to wake up and to help Tanjiro. She does something, because some flames appear but it's unclear what happens, what exactly did she do? How did she help Tanjiro exactly?
She uses her Blood Demon Technique - Exploding Blood to ignite her blood that was splattered over Rui's webs. This incinerates the threads that where about to cut Tanjirō and also ignites the blood on Tanjirō's blade which allows him to cut off Rui's head.