Ireland is quite connected to snakes, perhaps more than any other country in Europe. Serpents appear in many different tales and fill the folklore of Ireland. The possible answer to why this is could be that the Celts brought serpent cults with them, as the Celts' religion was broken up into different cults. The serpent, however, was not seen as solely an evil being. The creature was seen as both good and bad, for the venom of a snake could bring about medicine, yet the poison was deadly.
Before Christianity, there were some crosses within ancient Ireland. These crosses were often accompanied by serpents or dragons, in some cases. In addition, sculptures, glass, and fonts have turned up with serpents on them.
When Christianity came to Ireland, the religion changed the view of the serpent. In Christianity, serpents, snakes, and dragons are all related to the great evil, or the devil. For this, the saints of Ireland drove out all the snakes in Ireland.
Now, the likelihood of the saints driving out snakes literally is quite unreal. However, what the folk tales could be speaking of is the 'snake men' who worshipped serpents, snakes, and dragons. These people, as part of religious cults other than Christianity, would be unwelcome in Ireland after the coming of the new religion.
Some famous Draconis Extinctors include Saint Patrick, Saint Clement, Saint Columcille, and Saint Murrough. Saint Patrick drove out, among many different serpents, Ollipeist, one of the most famous Irish dragons. Saint Clement drove out the Dragon of Matz, while Saint Columcille used his followers to rid Ireland of the dragon from Donegal. Perhaps the most famous of these Draconis Extinctors, however, would be Saint Murrough, who drove out the last serpent of Ireland, Paiste.
In all of these stories, the reason for the dragon being driven out was due to the fact that the Saints were holy men and true believers of the Christian God.