I haven’t heard this one in a while, but I thought it was worth “digging up,” because it really is an “old saying.” Similar to the more modern “double down,” to dig one’s heels in means to refuse to do something such as change one’s plans, particularly to meet someone else’s demands to do so. The origin in this usage is illusive, however, it obviously comes from forcing the heels of one’s shoes into the ground and holding them down stubbornly, refusing to move.