This means that both sides of the debate/argument/etc. are unwilling to compromise or change their view. It comes from trench warfare, where battle lines changed incredibly slowly and it was almost impossible to gain any new ground.

The term originates from fortified trenches dug into the ground during military conflict. Many of the photos and films from World War I, for example, show the two opposing sides dug into trenches facing each other and neither side will budge or give up ground to surrender their position. It is used in English to describe attempted negotiations wherein two parties will not change their positions or agree to make suggested changes easily.