Usually, large force is employed on a relatively small portion of the front to achieve this. While the line may have held for a long while prior to the breakthrough, the breakthrough marks a relatively small time frame where the pressure on the defender leads him to "snap" in a very short time span.
As the first unit breaks, the adjacent units suffer adverse results from this (spreading panic, additional defensive angles, threat to supply lines) and, since they were already pressured as well, leads them to "snap" as well, causing a domino-style collapse of the defensive system. The defensive force thus evaporates at the breakthrough point, giving the attacker the option to rapidly move troops into the gap, exploiting the breakthrough in width (by attacking enemy units at the edge of the breakthrough, so widening it), in depth (advancing into enemy territory and strategic objectives), or a combination of both.
- Paddy Griffith (1994). Battle Tactics of the Western Front: The British Army's Art of Attack, 1916-18. Yale University Press. p. 32.