The problem it was built to solve
The away goals rule existed because playing a second leg away from home was, decades ago, a genuinely harder task: pitches varied enormously in condition, travel was more disruptive, and home advantage across two legs could feel close to decisive. Awarding extra weight to goals scored on the road was a way of nudging both teams toward attacking rather than defensive tactics in the away leg, since a team desperate merely to avoid defeat had less to gain from a goalless draw than one where an away goal broke the tie in its favour.
Why the imbalance faded
Over time, the conditions the rule was designed to correct changed substantially. Pitch quality became far more standardised, travel and preparation improved, and home advantage across two legs narrowed considerably compared with the era the rule was written for. As that imbalance shrank, the rule increasingly produced a different, less attractive incentive: teams playing the first leg at home began setting up defensively specifically to avoid conceding an away goal, since it now carried extra weight regardless of whether the underlying disadvantage of playing away had actually persisted.
A lesson in incentive design
Its eventual removal is a useful case study beyond football itself: a rule can be well designed for the conditions it was written in and still become counterproductive once those conditions change, because the behaviour it rewards adapts long before anyone revisits the rule. Competition organisers eventually concluded that the away goals rule was producing more defensive first legs than attacking second legs, the opposite of its original intent, and chose to treat all goals equally rather than keep patching a rule whose original justification had quietly disappeared.