Where in the Bible is the restoration of the nation of Israel prophesied? I have heard that the restoration of the state of Israel is a fulfillment of one or more Biblical prophecies. But I cannot find any explicit reference to it.

Is this really a fulfillment of prophecy, if so which one(s)?
In Ezekiel 37, the prophet has a vision of a valley of "dry bones."  When asked if these bones could yet live, Ezekiel wisely says, "You alone know, O Lord." (v3) After this, the Lord miraculously makes the bones come together and come alive.

It is explicitly stated in that chapter, that this refers to the people of Israel being re-established in the land of Israel:


  Then he said to me: “Son of man, these bones are the people of Israel. They say, ‘Our bones are dried up and our hope is gone; we are cut off.’  Therefore prophesy and say to them: ‘This is what the Sovereign Lord says: My people, I am going to open your graves and bring you up from them; I will bring you back to the land of Israel. - Ezekiel 37:11-12 NIV


Additional context is set in the previous chapter where Ezekiel, speaking for the Lord says:


  “‘But you, mountains of Israel, will produce branches and fruit for my people Israel, for they will soon come home. 9 I am concerned for you and will look on you with favor; you will be plowed and sown, 10 and I will cause many people to live on you—yes, all of Israel. The towns will be inhabited and the ruins rebuilt. 11 I will increase the number of people and animals living on you, and they will be fruitful and become numerous. I will settle people on you as in the past and will make you prosper more than before. Then you will know that I am the Lord. 12 I will cause people, my people Israel, to live on you. They will possess you, and you will be their inheritance; you will never again deprive them of their children. - Ezekiel 36:8-12 NIV


Hal Lindsey, (The Late Great Planet Earth), for instance, explains how he sees this as the restoration of Israel in this video Note: Hal Lindsey is a controversial figure, to say the least, but he is the direct inspiration for many premilennial dispensationalists, such as Tim LaHaye (Left Behind), John Walverood, Ed Whisenant, and others.

Ed Whisenant's well-known work 88 Reasons why the Rapture will occur in 1988 was fundamentally based on the conflation of this event, Jesus' parable of the fig tree (not a generation shall pass away until these things occur) and the restoration of Israel in 1948.