Question

Why does Goines say he wrote a letter to the draft board?

Answer 1

In “Let Sleeping Dogs Lie” David Goines describes how he avoided the military service during the Vietnam war, he wrote several letters to the draft board requesting them to be removed from the list for many reasons in order to not being recruited which worked until the war was ended, he did not agree with the military service and used his book also to protest against the draft board and the war.

Answer 2

Does this question have the following list of options to choose from?
to protest the war
to demonstrate his love of country
to prove his political activism
to ensure he would not get drafted

If so, then the answer would be the third one. Goines is using satire, a type of writing, making fun of human behavior  to condemn it or find some social change. He uses this type of writing so as to have the people/ reader think about the war more critically.