Potatoes were/are an important crop in Ireland. The speaker's father was a potato farmer. The speaker admires his father's work ethic and/or skill in using a shovel. The speaker also recalls his grandfather, another man very skillful with a shovel. He notes his grandfather could "cut more turf in a day / Than any other man in Toner's bog." Toner is presumably an Irish name. This is the land upon which his grandfather worked, passed down from previous generations. Or, it is the name of the family who owns the land. Irish people used (and some continue to use) turf as fuel instead of coal. His father and grandfather worked the land for food and fuel.
The history of turf and potatoes in Irish culture parallels the speaker's own family history. He speaks of his father and grandfather digging - down to the roots. This shows his appreciation for his own family roots. There is also the morbid association of digging with graves. They worked to support their families and this shows how their harsh work led to their own graves. His predecessors worked hard labor for a living. He, the speaker, admires them for this but chooses another path: writing. He intends to dig with the pen. By this, he means that he will make a living as a writer, but he will also dig up his roots and remember his Irish family heritage in writing.