A brief summary of Melisandre's unexpected arrival at Winterfell: she helpfully lights a bunch of stuff on fire with her magic; tells Ser Davos that he doesn't need to execute her since she'll die by the end of the night; gives Arya the pep talk she needs to kill the Night King (at least I think that's what their chat about eyes is all about); and then once the battle is over and the North is victorious, she takes off the magic red necklace that has kept her youthful all this time, walks out into the snowy remains of Winterfell, and rapidly ages before turning to dust as Davos looks on.