As you note, these words come from 1 Timothy 6:12. More specifically, these are the words of the apostle Paul to his beloved son in the faith, Timothy.
The main point of these words is that God has a race marked out for his people. Therefore, the people of God must run well. They need to fight the good fight of faith. And through this fighting well, they will inherit eternal life. We see that Paul actually does this, when he says later on in another book that he has fought the good fight and is now being poured up as a drink offering.
In light of his biblical insight, we can say that Sir Thomas More also needs to fight the good fight of faith. Henry VIII might be the king and he might have great power, but in the end More needs to be faithful to God and his conscience. Others might kowtow to save their own skin and to curry favor, but these people cannot be said to have fought the good fight. Similarly, the common man must also fight the good fight and do what is right.