Most Jewish people interpret this chapter as a continuation of the personification of the nation of Israel from the previous chapter. The Targum of Jonathan, which is a midrashic commentary in the form of a loose translation, explains this chapter as having two separate subjects. The first, receives all the exaltation, that is the Moshiac. The second, who suffers, is the nation of Israel.
Isaiah 52:13-53:12 (Targum of Johnathon)
52:13. Behold my servant Messiah shall prosper; he shall be high, and increase, and be exceeding strong
52:14. as the house of Israel looked to him during many days, because their countenance was darkened among the peoples, and their complexion beyond the sons of men,
52:15. so will he scatter many peoples; at him kings shall be silent, and put their hands upon their mouth, because that which was not told them have they seen, and that which they had not heard they have observed.
53:1. Who hath believed this our glad tidings? and the strength of the mighty arm of the Lord, upon whom as thus hath it been revealed?
53:2. The righteous will grow up before him, yeah, like blooming shoots, and like a tree which sends forth its roots to streams of water will they increase - a holy generation in the land that was in need of him; his countenance no profane countenance, and the terror at him not the terror at an ordinary man; his complexion shall be a holy complexion, and all who see him will look wistfully upon him.
53:3. Then he will become despised, and will cut off the glory of all the kingdoms; they will be prostrate and mourning, like a man of pains and like one destined for sicknesses; and as though the presence of the Shekhinah had been withdrawn from us, they will be despised, and esteemed not.
53:4. Then for our sins he will pray, and our iniquities will for his sake be forgiven, although we were accounted stricken, smitten from before the Lord, and afflicted.
53:5. But he will build up the Holy Place, which has been polluted for our sins, and delivered to the enemy for our iniquities; and by his instruction peace shall be increased upon us, and by devotion to his words, our sins will be forgiven us.
53:6. All we like sheep had been scattered, we had each wandered off on his own way; but it was the Lord's good pleasure to forgive the sins of all of us for his sake.
53:7. He prayed, and he was answered, and ere even he had opened his mouth he was accepted; the mighty of the peoples he will deliver up like a sheep to the slaughter and like a lamb dumb before her shearers; there shall be none before him opening his mouth or saying a word
53:8. Out of chastisements and punishment he will bring our captives near; the wondrous things done to us in his days who shall be able to tell? For he will cause the dominion of the Gentiles to pass away from the land of Israel and transfer to them the sins which my people have committed.
53:9. He will deliver the wicked into Gehinnom, and those that are rich in possessions into the death of utter destruction, in order that those who commit sin may not be established, nor speak deceits with their mouth.
53:10. But it is the Lord's good pleasure to try and to purify the remnant of his people, so as to cleanse their souls from sin; these shall look on the Kingdom of their Messiah, their sons and their daughters shall be multiplied, they shall prolong their days, and those who perform the Law of the Lord shall prosper in his good pleasure.
53:11. From the subjection of the nations he will deliver their souls, they shall look upon the punishment of those that hate them, and be satisfied with the spoil of their kings; by his wisdom he will hold the guiltless free from guilt, in order to bring many into subjection to the law; and for their sins he will intercede.
53:12. Then will I divide for him the spoil of many peoples, and the possessions of strong cities shall he divide as prey, because he delivered up his soul to death, and made the rebellious subject to the Law: he shall intercede for many sins, and the rebellious for his sake shall be forgiven.
Notice that while this interpretation does attribute much of this chapter to the Moshiac, it is more of a traditional Jewish version of the Moshiac. He is an exalted king, he returns the exiles, he causes the dominions of the Gentiles to cease, he rebuilds the temple, and he reinstates, keeps, and defends the law.
Later commentaries tend to drop the emphasis on the Moshiac, and attribute the whole of the chapter to Israel itself. These commentaries focus on the exile and then the redemption of Israel.