Why will the Antichrist persecute the saints?
The Antichrist persecutes the saints because he is possessed and empowered by the ancient spirit of antichrist that has always hated God's people, and because God allows this persecution as a purging, testing fire meant to purify and prove the faith of the saints so they may reign with Christ.
The spirit that will animate the Antichrist is the same spirit of antichrist that caused Jesus to be crucified and that later inspired the slaughter of the early church. That spirit denies that Jesus Christ has come in the flesh, and it hates everything that belongs to God. In the last days this spirit will be manifested fully in a man, the Antichrist, who will be given power to make war with the saints and overcome them for a season, holding authority over every kindred, tongue and nation.
Yet this persecution does not happen outside of God's rule. God rules in the kingdoms of men, and the Antichrist can only do what he is permitted to do. He is allowed to act as a buffeting spirit sent against the Church so that believers might suffer with Christ in order that they might also reign with Him. In the same way, he becomes a rod of correction against Israel for having rejected their Messiah.
The time of the Antichrist is described as a season of testing, purging and proving. It is when the Lord discovers whether His people will wait upon Him and remain steadfast in His Word despite affliction, imprisonment, and threat of death. Some of the wise and understanding among God's people will fall, but this is said to be in order to try them, to purge them, and to make them white, even to the appointed time of the end.
During this same period, iniquity abounds and the love of many grows cold, so that some believers themselves fall away, are offended, and even betray one another, adding to the suffering the saints endure. But those who endure to the end are promised salvation, and during this very persecution the saints become the light of the world, since their patience, faith and good works in the middle of such trial are meant to point others toward salvation while the Antichrist's cruelty is exposed for what it is.
The scriptures
A third part of thee shall die with the pestilence, and with famine shall they be consumed in the midst of thee: and a third part shall fall by the sword round about thee; and I will scatter a third part into all the winds, and I will draw out a sword after them.
And they worshipped the dragon which gave power unto the beast: and they worshipped the beast, saying, Who is like unto the beast? who is able to make war with him? And there was given unto him a mouth speaking great things and blasphemies; and power was given unto him to continue forty and two months. And he opened his mouth in blasphemy against God, to blaspheme his name, and his tabernacle, and them that dwell in heaven. And it was given unto him to make war with the saints, and to overcome them: and power was given him over all kindreds, and tongues, and nations. And all that dwell upon the earth shall worship him, whose names are not written in the book of life of the Lamb slain from the foundation of the world. If any man have an ear, let him hear. He that leadeth into captivity shall go into captivity: he that killeth with the sword must be killed with the sword. Here is the patience and the faith of the saints.
And such as do wickedly against the covenant shall he corrupt by flatteries: but the people that do know their God shall be strong, and do exploits. And they that understand among the people shall instruct many: yet they shall fall by the sword, and by flame, by captivity, and by spoil, many days. Now when they shall fall, they shall be holpen with a little help: but many shall cleave to them with flatteries. And some of them of understanding shall fall, to try them, and to purge, and to make them white, even to the time of the end: because it is yet for a time appointed.
And every spirit that confesseth not that Jesus Christ is come in the flesh is not of God: and this is that spirit of antichrist, whereof ye have heard that it should come; and even now already is it in the world.
While I was with them in the world, I kept them in thy name: those that thou gavest me I have kept, and none of them is lost, but the son of perdition; that the scripture might be fulfilled.
And after the sop Satan entered into him. Then said Jesus unto him, That thou doest, do quickly.
Then shall they deliver you up to be afflicted, and shall kill you: and ye shall be hated of all nations for my name’s sake. And then shall many be offended, and shall betray one another, and shall hate one another. And many false prophets shall rise, and shall deceive many. And because iniquity shall abound, the love of many shall wax cold. But he that shall endure unto the end, the same shall be saved. And this gospel of the kingdom shall be preached in all the world for a witness unto all nations; and then shall the end come.
Now we beseech you, brethren, by the coming of our Lord Jesus Christ, and by our gathering together unto him, That ye be not soon shaken in mind, or be troubled, neither by spirit, nor by word, nor by letter as from us, as that the day of Christ is at hand. Let no man deceive you by any means: for that day shall not come, except there come a falling away first, and that man of sin be revealed, the son of perdition; Who opposeth and exalteth himself above all that is called God, or that is worshipped; so that he as God sitteth in the temple of God, shewing himself that he is God.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.