Nezuko is Tanjiro's sister. Unfortunately, one day while Tanjiro was out getting wood, demons came and slaughtered her and her family. Instead of dying though, she was turned into a demon. But, there still seems to be a remanence of the old Nezuko left. In this scene, Tanjiro is trying to get her to safety and to see if anyone could treat her when a Demon Slayer by the name of Giyuu comes and interferes with Tanjiro's plan. He tries to kill Nezuko until he sees for himself how just because she's now a demon, it doesn't mean she can't protect the one person whom she loves. 

Here's a video summarizing the whole scene: https://www.youtube.com/watch?v=47eDlsbTnBg&t=70s