Why does the audience traditionally stand during the Hallelujah Chorus?

Because when King George the second heard it, he was so inspired, that he stood up. When everyone saw him stand, they stood too. In has been a tradition ever since then to stand when the Messiah is played.

The Hallelujah Chorus, is a Song of Praise - literally the Word "Hallelujah" meaning "Praise be to G-d". The text of Handel's great piece comes from the Old Testament Psalms (as requoted in the New Testament Book of Revelations), Words which continue to be sung and recited in their original Hebrew form in synagogues to this day during a part of the service known as "Hallel". It has always been the custom for Jews to stand whilst praising the Lord during "Hallel", a practise which was passed on to Christendom where it is customary for congregations to stand for Songs of Praise (or Hymns). As Head of the Church of England, King George would certainly have been aware of the obligation to stand for "Hallelujah" - and might well have remained standing until the last AMEN at the end of the movement.