Why does the Green Knight challenge Sir Gawain?
The Medieval tale of Sir Gawain and the Green Knight shows the importance of chivalry and honor.
The Green Knight, a pawn of Morgan Le Fay, challenges the Green Knight so as to play out the game which Le Fay has created so as to scare Guinevere.
Sir Gawain is a member of Arthur's Court. As a member of his court, he is required to uphold the code of the knight. When the Green Knight comes to Arthur's, during the Christmas season, he offers a challenge. Arthur, not willing to eat until entertained, allows the Green Knight to put forth his challenge. None of the knights step forward at first. Arthur, somewhat ashamed of his knights, initially accepts the Green Knight's challenge.
Gawain, knowing the he may fail and seeing himself as the weakest on one who will be least missed, decides to take Arthur's place. It is out of courtly love and chivalric duty by which he does this.
Le Fay's challenge, again as enacted by the Green Knight, is a simple ruse because of her hatred and jealousy against Queen Guinevere. Her whole plan is to frighten Guinevere so as to gain an upper hand over the Queen.