Isaiah 52:13-53:12 constitutes what is considered the fourth Servant song. The others are Isaiah 42:1-9, Isaiah 49:1-13, and Isaiah 50:4-9. Also, some consider Isaiah 61:1-3 a fifth Servant song, though the word "servant" is not used there.
All of these songs speak of a Servant called by God to lead the nations. There is no clear referent within Isaiah itself as to the identity of this Servant. The identity of the Servant is even said to be hidden at points (Isaiah 49:2). For this reason, several interpretations have emerged.
One traditional Jewish interpretation is that the Servant is a metaphor for the nation of Israel. This interpretation is especially shaped by Isaiah 49:3 - "And he said to me, 'You are my servant, Israel, in whom I will be glorified.'" One problem with this interpretation, though, is raised by verses 49:5-6. There we see that the Servant is formed to bring Israel back to the LORD. This servant Israel therefore seems separate from the nation Israel.
Others within even the Jewish tradition have therefore interpreted the Servant as an individual, possibly a future messianic figure. One can make sense of 49:3 in this interpretation by seeing that the name Israel was originally given to an individual (Genesis 35:10). Various individuals have been put forth as this figure: e.g. Moses, Hezekiah, Zerubbabel, or Cyrus.
Most of these interpretations center around a figure who lead Israel back into the land. However, one of the characteristics of this Servant is that he not only leads Israel back, but that he is a light and salvation to the nations and that he leads them back not necessarily to the land, but to God himself (cf. 49:6, 52:15).
Christians therefore believe that Jesus is the fulfillment of these Servant songs. One thinks of Athanasius's observation that "though there is no longer any king or prophet nor Jerusalem nor sacrifice nor vision among [the Jews]; yet the whole earth is filled with the knowledge of God, and the Gentiles, forsaking atheism, are now taking refuge with the God of Abraham through the Word, our Lord Jesus Christ."
What Isaiah imagined writing the words in these songs cannot be certain. But I don't think it is a stretch to say that he imagined some messianic figure.