"No man can serve two masters" is taken from Matthew 6:24 and Luke 16:13. Jesus was explaining that it is not possible to devote yourself to God and to any other power or pursuit at the same time: "for you will love the one and hate the other." Henry VIII had been declared "Defender of the Faith" by the pope for a treatise he wrote in defense of church doctrine. By quoting from the New Testament, More is reminding the king that he serves God above all and will serve no other master, not even the king of England.
By this More means he cannot acquiesce to Henry's demands that he speak in support of his divorce and marriage as well as be true to his conscience, which he understands in terms of the tenets of the Catholic Church's position concerning the invalidity of divorce. While More struggles mightily to be give loyalty to King Henry and do what his conscience demands, in the end, they compete, and he realizes he must choose who his ultimate master is. He chooses his conscience and God.