It is important to realize who the apostle Paul is writing about in this verse. The antecedent of "they" appears in the previous verse: "those who are in opposition." The entire epistle is instruction for the evangelist Timothy, and in this passage in particular, Paul is giving the younger man advice on how to handle those who dispute the gospel message he taught.
He instructs Timothy, as "a servant of the Lord," to correct his opponents with humility and in the hope of two positive outcomes should God grant repentance to them. First, his correct explanation of the matter in contention would bring them out of their ignorance, liberating them from the bondage of error (John 8:32) and opening the potentialities of the truth to them. Paul was very aware that false teachers and anti-Christian foes functioned with a veil over their minds (see, for instance, how he explains it regarding the Jews in II Corinthians 3:14-16; Matthew 15:14), a blindness that could only be lifted by the direct intervention of God revealing Himself and His truth by the Holy Spirit (I Corinthians 2:10-14; John 6:44). A minister of God should always answer naysayers plainly with the revealed truth of God to give them the knowledge that may lead to their repentance.
The second positive outcome is the subject of II Timothy 2:26. He hopes that exposure to the truth will bring opponents "to their senses" and free them from their captivity to Satan. The apostle realizes that even the most cunning argument of one of God's servants is not enough to accomplish this; a person's repentance and acceptance of the truth will happen only if God "flips a switch" in his mind by the Holy Spirit to become receptive to Him. So a minister must present the truth in the event that God will use his explanation to call him into a relationship with Him. It is only at this point that an individual truly comes to his senses (see Luke 15:17; Acts 9:3-20). Only then does he begin to see without the blinders (or in Paul's own case, when the scales fell from his eyes).
Once one accepts the sacrifice of Jesus Christ and acknowledges Him as personal Savior and Lord, the walls of Satan's prison fall away and crumble to dust (see Romans 6:16-22). His power over us disappears because his claim on us has been removed; our sins have been forgiven and we are no longer in rebellion against God. We have gone over to the other side in the great spiritual war that the Devil has always been destined to lose. The Captain of our salvation has already crushed the head of the serpent (Genesis 3:15), and all that remains is the perfection of the saints for their roles in the Kingdom of God.
However, there are yet billions of people who are still "captive . . . to do his will." Revelation 12:9 states that the great dragon, who is the Devil and Satan, has deceived the whole world. Despite his ignominious defeat at Calvary, Satan is determined to turn it into victory. In his pride, he still thinks he can win! So he will continue to oppose God and His people wherever and whenever he can, using his captives all over the world to trouble, persecute, and kill God's saints. This reality means that Christians must remain on their guard at all times, prepared to "fight the good fight" (II Timothy 4:7) to wear the crown of victory in the Kingdom.
Finally, we must remember that our fight is really not against the men and women still enslaved to Satan, although they are the faces and voices that oppose us. Paul writes:
For we do not wrestle against flesh and blood, but against principalities, against powers, against the rulers of the darkness of this age, against spiritual hosts of wickedness in the heavenly places. Therefore take up the whole armor of God, that you may be able to withstand in the evil day, and having done all, to stand. (Ephesians 6:12-13)
We need to look beyond our physical opponents to the evil spirits behind them, realizing that our human foes have not yet come to their senses and seen the light of the truth that only God can reveal. Thus, we can contend with them in humility and gentleness, grateful for the grace God has extended to us.
