Because that for his name's sake - The word "his" here refers to God; and the idea is, that they had undertaken this journey not on their own account, but in the cause of religion.
They went forth - Or, "they have gone forth" - ἐξῆλθον exēlthon - referring to the journey which they had then undertaken; not to the former one.
Taking nothing of the Gentiles - The term "Gentile" embraced all who were not "Jews," and it is evident that these persons went forth particularly to labor among the pagan. When they went, they resolved, it seems, to receive no part of their support from them, but to depend upon the aid of their Christian brethren, and, hence, they were at first commended to the church of which Gaius and Diotrephes were members, and on this second excursion were commended particularly to Gaius. Why they, resolved to take nothing of the Gentiles is not stated, but it was doubtless from prudential considerations, lest it should hinder their success among them, and expose them to the charge of being actuated by a mercenary spirit. There were circumstances in the early propagation of Christianity which made it proper, in order to avoid this reproach, to preach the gospel "without charge," those to whom it is preached to contribute to its maintenance, and that it is the right of those who preach to expect and receive a support. On this subject, see the 1 Cor. 9 notes, particularly 1-John 1:15, 1-John 1:18 notes.

For his name's sake they went forth - For the sake of preaching the Gospel of the grace of God, and making known Jesus to the heathen.
Taking nothing of the Gentiles - Receiving no emolument for their labor, but in every respect showing themselves to be truly disinterested. Sometimes, and on some special occasions, this may be necessary; but the laborer is worthy of his hire is the maxim of the author of Christianity. And those congregations of Christians are ever found to prize the Gospel most, and profit most by it, who bear all expenses incident to it, and vice versa.
But some construe εξηλθον, they went out, with απο των εθνων, from the Gentiles, or rather by the Gentiles, and give the passage this sense: They went out, i.e., were driven out by the Gentiles, taking nothing with them, i.e., leaving all their property behind, so that they were in a state of great destitution. A curious reading here, εθνικων, heathenish men, for εθνων, Gentiles, which latter might imply those who were converted from among the Gentiles, while the sense of the other term seems to be restrained to those who were still unconverted, may seem to strengthen the above interpretation; and although the construction seems rather harsh, yet it is not, on the whole, unlikely. The reading above referred to is that of the most ancient and reputable MSS. That to be driven out or expelled is one scriptural meaning of the verb εξερχομαι, see Matthew 8:32 : And when they were come out, οἱ δε εξελθοντεσς, and when they were Driven Out. Matthew 12:43 : When the unclean spirit is gone out, εξελθη, is Driven Out. See Mark 5:13, Mark 7:29 : The devil is gone out of thy daughter, εξεληλυθε, is Expelled. Mark 9:29 : This kind can come forth by nothing εν ουδενι δυναται εξελθειν, can be Driven Out by nothing, but by prayer and fasting. Luke 8:2 : Mary Magdalene; out of whom went, αφ' ἡς δαιμονια ἑπτα εξεληλυθει, out of whom were Cast, seven demons. See also 1-John 2:19; Revelation 3:12; and Schleusner, in voc. εξερχομαι.

Because that for his name's sake they went forth,.... From Judea; either of their own accord to preach the Gospel, or being drove out by the unbelieving Jews, for professing the name of Christ; and be it which it will, there was good reason why they should be regarded, and especially since they did as follows,
taking nothing of the Gentiles; even of those who were converted, though their preaching the Gospel, to whom they ministered, for of others, the unconverted Gentiles, they could not expect to receive; and this they did, as the apostles before them, because they would not be chargeable to them, and lest it should be thought they sought their own worldly interest, and not the good of souls and glory of Christ, and so a stumblingblock be laid in the way of the Gospel, to hinder the progress of it. The Ethiopic version reads this in the singular number, "and I went forth for his name's sake, taking nothing of the Gentiles".

his name's sake--Christ's.
went forth--as missionaries.
taking nothing--refusing to receive aught by way of pay, or maintenance, though justly entitled to it, as Paul at Corinth and at Thessalonica.
Gentiles--the Christians just gathered out by their labors from among the heathen. As Gaius himself was a Gentile convert, "the Gentiles" here must mean the converts just made from the heathen, the Gentiles to whom they had gone forth. It would have been inexpedient to have taken aught (the Greek "meden" implies, not that they got nothing, though they had desired it, but that it was of their own choice they took nothing) from the infant churches among the heathen: the case was different in receiving hospitality from Gaius.

They went forth - To preach the gospel.