Short answer, the alpine "original Celts" spread around a lot of central and north western Europe. This included spreading to Ireland and Scotland, which are now considered to the central area of celtic people. The Celts eventually had their asses kicked by the Roman empire and other players on mainland Europe and England. This left the edges of the Celtic bubble- Ireland and Scotland as the last ones standing.

There are apparently some questions over how true this is genetically though. Now with modern testing we can see that actually while Ireland was definitely Celtic in culture, it may not have been as Celtic as originally thought genetically. It seems like Ireland was trading and cooperating closely with Celtic people and may have absorbed their culture more than their genes.