The story
From firmly placing one’s heels in the ground, particularly in tug of war (pulling) or when bracing oneself (pushing). Compare drag one's feet (“act slowly, from lack of enthusiasm or to delay”).
Etymology from Wiktionary, CC BY-SA 4.0.
dig in one's heels
Today it means: To act in a determined manner by firmly maintaining one's beliefs, demands, situation, etc. in the face of opposition or cajoling. Full definition →