The message of Isaiah rebounds back and forth between visions of judgment and glimpses of Messianic hope and renewal. Elsewhere in the prophets, the term "branch of the LORD" is a Messianic title (Jer 23:4; Jer 33:15; Zech 3:8; Zech 6:12). It is the meaning in this context as well. There is a glimmer of hope after the fall of Jerusalem and the suffering of the residents of Judah. Out of these ashes, God speaks of a glorious future hope for the nation. A mighty deliverer will be the hope of all the people. Beauty, glory, fruitful economic flourishing, and national pride will all return with "the branch of the LORD".
Messianic hope is a big deal in the major prophets. It is not a message hidden in the words of these holy men. It is revealed clearly. Christians are not reading anything into these passages that Jews have not already known in the prophets and held as their hope ever since the Babylonian Captivity. Israel's hope is for Messiah to come. It was what Isaiah longed for. And of all the prophets, he will foretell in detail the Messiah's coming glory.
Isaiah's Messianic hopes are often quoted in the New Testament. The earliest Christians (most were Jewish) saw in Jesus the clear fulfillment of them. And they made the connection because it was legitimate. In Isaiah we see Messiah as deliverer (from much more than just political oppression) and as sufferer. Jesus came clearly to do both. He suffered to deliver us from sin's dominion. He will come again to deliver Israel and restore the grandeur of the covenant with His people.