Why did Jesus say the dead girl was "not dead but asleep"?

Saint Theophylact of Ohrid answers ("The Explanation of the Holy Gospel according to St. Matthew", chapter 9):

Matthew 9:23-24. And when Jesus came into the ruler’s house, and saw the flute players and the people making a noise, He said unto them, Give place: for the maid is not dead, but sleepeth. And they laughed Him to scorn. Since she was unmarried, they were mourning her with flutes used at weddings, which was contrary to the law. Jesus said that she was sleeping, because He was able to resurrect her easily, and so to Him, death was sleep. Do not marvel that they laughed Him to scorn, for by scoffing they bear witness all the more to the miracle that He resurrected one who was truly dead. So that no one could later say that she had only suffered a seizure, it was confessed by all those present that she was dead.

Another, more extensive interpretation we can read in the "Homilies on the Gospel according to St. Matthew" of the Saint John Chrysostom (homily 31, parts 3-4):

Noble tokens, surely, these, of the rulers of synagogues; in the moment of her death pipes and cymbals raising a dirge! What then doth Christ? All the rest He cast out, but the parents He brought in; to leave no room for saying that He healed her in any other way. And before her resurrection too, He raises her in His word; saying, “The maid is not dead, but sleepeth.” And in many instances besides He doeth this. As then on the sea He expels tumult from the mind of the bystanders, at the same time both signifying that it is easy for Him to raise the dead (which same thing He did with respect to Lazarus also, saying, “Our friend Lazarus sleepeth”) and also teaching us not to fear death; for that it is not death, but is henceforth become a sleep. Thus, since He Himself was to die, He doth in the persons of others prepare His disciples beforehand to be of good courage, and to bear the end meekly. Since in truth, when He had come, death was from that time forward a sleep. But yet they laughed Him to scorn: He however was not indignant at being disbelieved by those for whom He was a little afterwards to work miracles; neither did He rebuke their laughter, in order that both it and the pipes, and the cymbals, and all the other things, might be a sure proof of her death.

For since for the most raft, after the miracles are done, men disbelieve, He takes them beforehand by their own answers; which was done in. the case both of Lazarus and of Moses. For to Moses first He saith, “What is that in thine hand?” in order that when he saw it become a serpent, He should not forget that it was a rod before, but being reminded of his own saying, might be amazed at what was done. And with regard to Lazarus He saith, “Where have ye laid him?” that they who had said, “Come and see,” and “he stinketh, for he hath been dead four days,” might no longer be able to disbelieve His having raised a dead man.

Seeing then the cymbals and the multitude, He put them all out, and in the presence of the parents works the miracle; not introducing another soul, but recalling the same that had gone out, and awakening her as it were out of a sleep. And He holds her by the hand, assuring the beholders; so as by that sight to make a way for the belief of her resurrection. For whereas the father said, “Lay thy hand upon her;” He on His part doth somewhat more, for He lays no hand on her, but rather takes hold of her, and raises her, implying that to Him all things are ready. And He not only raises her up, but also commands to give her meat, that the event might not seem to be an illusion. And He doth not give it Himself, but commands them; as also with regard to Lazarus He said, “Loose him, and let him go,” and afterwards makes him partaker of His table. For so is He wont always to establish both points, making out with all completeness the demonstration alike of the death and of the resurrection.

But do thou mark, I pray thee, not her resurrection only, but also His commanding “to tell no man;” and by all learn thou this especially, His freedom from haughtiness and vainglory. And withal learn this other thing also, that He cast them that were beating themselves out of the house, and declared them unworthy of such a sight; and do not thou go out with the minstrels, but remain with Peter, and John, and James. For if He cast them out then, much more now. For then it was not yet manifest that death was become a sleep, but now this is clearer than the very sun itself. But is it that He hath not raised thy daughter now? But surely He will raise her, and with more abundant glory. For that damsel, when she had risen, died again; but thy child, if she rise again, abides thenceforth in immortal being. Let no man therefore beat himself any more, nor wail, neither disparage Christ’s achievement. For indeed He overcame death.

Why then dost thou wail for naught? The thing is become a sleep. Why lament and weep? Why, even if Greeks did this, they should be laughed to scorn; but when the believer behaves himself unseemly in these things, what plea hath he? What excuse will there be for them that are guilty of such folly, and this, after so long a time, and so clear proof of the resurrection? But thou, as though laboring to add to the charge against thee, dost also bring us in heathen women singing dirges, to kindle thy feelings, and to stir up the furnace thoroughly: and thou hearkenest not to Paul, saying, “What concord hath Christ with Belial? or what part hath he that believeth with an infidel?” And while the children of heathens, who know nothing of resurrection, do yet find words of consolation, saying, “Bear it manfully, for it is not possible to undo what hath taken place, nor to amend it by lamentations;” art not thou, who hearest sayings wiser and better than these, ashamed to behave thyself more unseemly than they? For we say not at all, “Bear it manfully, because it is not possible to undo what hath taken place,” but, “bear it manfully, because he will surely rise again;” the child sleeps and is not dead; he is at rest and hath not perished. For resurrection will be his final lot, and eternal life, and immortality, and an angel’s portion. Hearest thou not the Psalm that saith, “Return unto thy rest, O my soul, for the Lord hath dealt bountifully with thee?” God calleth it “bountiful dealing,” and dost thou make lamentation? And what more couldest thou have done, if thou wert a foe and an enemy of the dead? Why, if there must be mourning, it is the devil that ought to mourn. He may beat himself, he may wail, at our journeying to greater blessings. This lamentation becomes his wickedness, not thee, who art going to be crowned and to rest. Yea, for death is a fair haven. Consider, at any rate, with how many evils our present life is filled; reflect how often thou thyself hast cursed our present life. For indeed things go on to worse, and from the very beginning thou wert involved in no small condemnation. For, saith He, “In sorrow that shalt bring forth children;” and, “In the sweat of thy face shalt thou eat thy bread;” and, “In the world ye shall have tribulation.” But of our state there, no such word at all is spoken, but all the contrary; that “grief and sorrow and sighing have fled away.” And that “men shall come from the east and from the west, and shall recline in the bosoms of Abraham and Isaac and Jacob.” And that the region there is a spiritual bride-chamber, and bright lamps, and a translation to Heaven.