J. J. M. Roberts provides an explanation of Second Isaiah.
J. J. M. Roberts, "Introduction to Isaiah," HarperCollins Study Bible, ed. by Harold Atrridge (San Fransisco: HarperCollins Publishers, 2006), 913
Chapters 40-55 presuppose an Israelite audience living in Babylon toward the end of the Babylonian exile (597-539 BCE). The prophet announces to his listeners that the end of their exile in Babylon is imminent. Babylon, not Assyria, is Israel's main enemy, and the burden of the prophet's message is the promise of deliverance, not the threat of judgment. Moreover, the prophet twice mentions the Persian ruler Cyrus (44.28; 45.1) as a figure who has come to the attention of his audience. Such notice certainly presupposes Cyrus's dethroning of his Median overlord Astyages (550) and perhaps Cyrus's defeat of the Lydian king Croesus (547/6) as well. Thus this anonymous prophet's work can probably be dated between 545 and 539 BCE.
Though clearly dependent on First Isaiah's message, Second Isaiah is also easily distinguishable. Second Isaiah shares the eighth-century prophet's emphasis on the holiness of God and his vision of God as the great king, and he at least adapts the earlier prophet's ideal of the Davidic king, democratizing it to apply it to the entire nation. Unlike First Isaiah, however, the message of Second Isaiah is primarily that of consolation, and this message is couched in a distinctive style. In relatively long, markedly lyrical oracles, Second Isaiah reassures the exile that God still controls history. Despite present appearances, the Lord will soon demonstrate his power by bringing the Israelites back to their own country in a second exodus more glorious than the first. Before this God, who has created all things, the Babylonian idols are as nothing; God's judgment on both the Babylonians and their gods is imminent.