Question: "What is meant by the first part of "I pray not that thou shouldest take them out of the world, but that thou shouldest keep them from the evil." in John 17:15? St. Theophylact is rather mute on that point."
To understand this verse, you have to read it in the context of the entire chapter, which is composed of what is often called Christ's "High Priestly Prayer." This prayer has three sections. It begins with a prayer for Christ Himself (1-5), then for His disciples (6-19), and finally for all who would come to believe in Him through His disciples (20-26).
Most relevant to the meaning of the verse in question is a theme throughout this prayer of "the world". The word "world" (kosmos) in the New Testament can mean the creation, the people of the world (such as in John 3:16: "for God so loved the world..."), and it can also refer to the fallen evil system of this world. We see all three meanings used in this prayer.
Christ has glorified the Father on earth (4) and now prays that the Father will glorify Him "with the glory which I had with Thee before the world was" (5). Christ has manifest the name of the Father "unto the men which Thou gavest me out of the world" (6). Christ in this prayer, prayed "not for the world, but for them which Thou hast given me..." (9). Christ, who now is preparing for His passion which would soon be upon Him says "And now I am no more in the world, but these [His disciples] are in the world..." (11), While He "was with them in the world," He kept them in the Father's name, and none were lost save Judas "that the scripture might be fulfilled" (12). And of His disciples He says "the world hath hated them, because they are not of the world, even as I am not of the world" (14). Then we have the verse in question: "I pray not that Thou shouldest take them out of the world, but that Thou shouldest keep them from the evil" (15). Which is immediately followed by the statement: "They are not of the world, even as I am not of the world" (16). Even as the Father "hast sent me into the world, even so have I also sent them into the world" (17).And He prays for all those who would come to believe, that they would be one, " that the world may know that Thou hast sent me..." (23), and that they may "be with me where I am; that they may behold my glory, which Thou hast given me: for Thou lovedst me before the foundation of the world" (24).
And so the point here is that Christ was sent into this world, and is about to complete His work, but just as He was sent into the world, His disciples are being sent into the world, that the world may know Him. And so now is not the time for them to be taken out of this world. Their work has only begun. They are in the world, but are not of it. And that is our calling, to fulfill the work that Christ has given us to do in this world, but to keep ourselves from becoming part of the evil that is in this world.
Here are what two Church Fathers have to say about this verse in particular:
St. Cyril of Alexandria says:
"What, then, is His prayer, after that He has shown that the disciples are hated by those who are fast bound by the evil things of the world? I pray not, He saith, that Thou shouldest take them from the world, but that Thou shouldest keep them from the evil one. For Christ does not wish them to be quit of human affairs, or to be rid of life in the body, when they have not yet finished the course of their apostleship, or distinguished themselves by the virtues of a godly life; but he wishes them, after they have lived their lives in the company of men in the world, and have guided the footsteps of those who are His to a state of life well pleasing to God, then at last, with the glory they have achieved, to be carried into the heavenly city, and to dwell with the company of the holy angels. We find, moreover, one of the Saints approaching the God Who loves virtue with the cry: Take me not away in the midst of my days; for pious souls cannot, without a pang, put off the garment of the flesh before they have perfected their life in holiness above their fellows. Therefore also the Law of Moses, teaching us that sinners are visited as in wrath, and by way of penalty, with premature death, often reiterates the warning to stand aloof from evil, that thou diest not before thy time. Besides, if the Saints chose to keep themselves apart from our daily life, it would infer no small loss to those who are unstable in the faith; nay, they could in nowise be guided in the way of righteousness, without the aid of those who are able to lead them therein. Paul knew this when he said, To depart and be with Christ is far better for me, yet to abide in the flesh is more needful for your sake. Christ, therefore, in His care for the salvation of the uninstructed, says that those who are in the world ought not to be left desolate without the Saints, who are men of light, and the salt of the earth; but prays rather for the safe keeping of His holy ones, and that they may be ever untouched by the malice of the evil one, shunning the assault of temptations by the power of His Omnipotent Father" (Commentary on John 11:9).St. John Chrysostom says:
"...Christ came not to put us to death and deliver us from the present life in that sense, but to leave us in the world, and prepare us for a worthy participation of our heavenly abode. Wherefore He saith to the Father, “And these are in the world, and I come to Thee; I pray not that Thou shouldest take them from the world, but that Thou shouldest keep them from the evil,” (John 17:11,15) i.e., from sin"(Homilies of Galatians 1:4).For more information on this chapter you can read three homilies St. John Chrysostom gave on this chapter:
Homily 80 on John 17:1-5
Homily 81 on John 17:6-13
Homily 82 on John 17:14-26