In 2 King 2 when the Lord was about to take Elijah up to heaven in a whirlwind, Elisha who was successor of Elijah’s prophetic ministry, stayed with him from Gilgal to Bethel and then to Jericho.  Elijah was preventing Elisha from following him because God called him to move forward from one place to another.  Elisha refused to leave his master Elijah and insisted on receiving a double potion of his spirit before Elijah was to be taken away.  Elijah said to Elisha in reply, “If you see me taken up, your wish will be granted.”  As they kept on walking and conversing, a chariot of fire and horses took up Elijah in a whirlwind.  Elisha saw it happen and picked up Elijah’s mantle which had fallen from him.  So the mantle is alluded to the passing of Elijah’s cloak or mantle to Elisha.  To claim the mantle is to claim the responsibility.  Example:

Peter has claimed the mantle of his party after a fierce competition.  Or

Tom takes up his father’s mantle to become the new president.