Because that for his name's sake they went forth, taking nothing of the Gentiles. His name's sake - Christ's.
Went forth - as missionaries. Taking nothing - refusing to receive anything as pay, or maintenance, though justly entitled to it: as Paul at Corinth and at Thessalonica (1 Corinthians 9:12; 1 Corinthians 9:15; 1 Thessalonians 2:6; 1 Thessalonians 2:9).
Gentiles. As Caius was a Gentile convert, "the Gentiles" must mean the converts just made from the pagan, the Gentiles to whom they had gone forth. It would have been inexpedient to have taken anything х meeden (G3367) implies, not that they got nothing, though they desired it; but, of their own choice, they took nothing from the infant Gentile churches]: the case was different in receiving hospitality from Caius.