Tom's discussion of writing early in the novel relates to two ideas primarily - humility and a lack of blame.
Tom wants to communicate that he knows how to write (pride) but that he does not like it and his father doesn't like it either. This is a way to claim a humble social position and to deny any ideas that he may be "putting on airs" or striving for false pretense. He wants to assert his basic abilities without suggesting anything unbecoming about himself or his family.
Also, Tom does not harbor any blame for the fact that his father has not written to him. Though this results in Tom arriving at the family home to find it empty, he holds no grudge against his father for failing to write a letter about the family's whereabouts.