The Wikipedia article notes that this was the case for many decades, but doesn't explain why. It also says that Shannon was a convenient place to refuel on flights between Europe and the US. But once airliners got long enough ranges to be able to fly directly to Dublin Airport, why weren't they permitted to do so until very recently?
This might be a technical issue, Shannon has a ~3200m runway while the longest one at Dublin is ~2600m. Glancing at this list of runway requirements it seems many longer ranged aircraft (such as those that cross the Atlantic) require at least 2700-2800m or so runway lengths.
It was Government policy for a 1:1 transatlantic flight ratio for Dublin and Shannon. The idea was to ensure business to what would have been seen as a "poorer" area. If an Airline wanted to fly from Ireland there had to be a Shannon stop-over. It was a long time before it stopped and a major political issue as local politicians wanted to keep the stop-over as it meant jobs etc.