Ezekiel's vision
In Ezekiel 4:3-6, God has a rather strange commandment for Ezekiel. God tells him "Lie down on your left side and stay there for three hundred ninety days as a sign of Israel's punishment - one day for each year of its suffering. Then turn over and lie on your right side forty more days. that will be a sign of Judah's punishment - one day for each year of it's suffering.". Mr. Jeffrey uses this number (430 years), minus time Israel and Judah have already paid in captivity, to calculate the date of 1948.
Israel's Captivity in Babylon
After Israel had entered the Promised Land (after leaving Egypt and wandering in the desert for 40 years) and after king Solomon had died, the people and kings rebelled against God and ignored His warnings to turn back to Him (Jeremiah 25:1-12). So as punishment, God had the king of Assyria attack Israel and capture them. They took the people of Israel back to Babylon to be servants (this is how Daniel, Shadrach, Meshach, and Abednego ended up serving king Nebuchadnezzar as described in the book of Daniel). God promised that this captivity would only last 70 years, as stated in Jeremiah 25:11 "This country will be as empty as a desert, because I will make all of you the slaves of the king of Babylonia for seventy years.".
This happened in the Jewish month of Nisan in 606 BC, and ended in Nisan in 536 BC [1, p.38], which is a total of 70 years exactly as promised. At this time, king Cyrus let the Jews leave Babylon and go back to Israel (Ezra 1:3). But in Ezra it describes how only a small number of Jews (42,360) actually return to Israel (see Ezra 2:64-67). Ezra then discovers that although the Jews were rebuilding the Temple, they hadn't really stopped their sinful ways. In Ezra 9:10, he says "Our God, what can we say now? Even after all this, we have disobeyed the commands that were given to us by your servants the prophets. They said the land you are giving us is full of sinful and wicked people, who never stop doing disgusting things.". Mr. Jeffrey states that only a small part of the Jews who returned to Israel repented of their sins, but the rest did not and a large majority stayed in Babylon [1, p.40].
So how do you get 1948?
In Leviticus 26, God specifies what sort of punishment there would be for disobeying Him. God says "If you disobey me and my laws, and if you break our agreement, I will punish you terribly..." (Leviticus 26:14-15, CEV) and proceeds to describe the types of punishments. Then God says "Then, if you still refuse to obey me, I will punish you seven times for each of your sins, until your pride is completely crushed." (Leviticus 26:18-19, CEV). Following this, again if there is no repentance then things get even worse - another 7 times increase and more and more disasters.
Mr. Jeffrey applies this principle to the number of years of punishment that remain according to Ezekiel's vision, minus the 70 years in Babylon. This gives a total of 430 - 70 = 360 years of punishment left. But he says that because the Jews did not fully repent when God let them come back to Israel, this remaining 360 years was then multiplied by 7 to give 2520 years (note that this is in "Biblical years", where one month is exactly 30 days and a year is exactly 360 days, not like our modern calendar of 365.25 days/year).
To calculate this in our modern frame of reference, Mr. Jeffrey converts these 2520 Biblical years to 2483.8 of our calendar years. This amount of time is added onto the year 536BC (the time when the Jews returned to Israel but did not repent), which produces a year of 1948. The state of Israel was officially declared on May 14, 1948. Mr. Jeffrey states that this is the first time since Solomon that the state of Israel has been an independent sovereign state [1, p.41]. This is true, because throughout the rest of Israel's history, Israel was always ruled as part of another empire [2] or some other country has had power over it.
Why does it matter?
First, this indicates that God still keeps his promises to Israel, even after all these centuries. If he remembers his promises to Israel, he will also his promises to us for our salvation.
Second, there is a very popular verse relating the rebirth of Israel to when Jesus returns to set up the Millennial kingdom. This verse is Matthew 24:32-34 where Jesus tells his disciples: "Learn a lesson from a fig tree. When it's branches sprout and start putting out leaves, you know that summer is near. So when you see all these things happening, you will know that the time has almost come. I can promise you that some of the people of this generation will still be alive when all this happens.".
The fig tree was commonly used as a symbol for Israel in the Bible [3]. Many scholars interpret this to mean that the same generation who witnessed the rebirth of Israel will still be alive when Jesus returns after the Tribulation to set up his kingdom. Some people reject this and say that when Jesus said "this generation", he was referring to those people alive when Jesus was alive, but since they have all died that therefore Jesus was wrong. This is untrue, as it seems clear in the verse that Jesus is referring to the generation alive when the fig tree sprouts (when Israel is reborn).
Currently, 1948 was approximately 63 years ago. Most people on average (at least in developed countries) seem to live to be an average of 80 or so years. So that may indicate that sometime in the next 17 years the Tribulation will happen and Jesus will return. Since the Tribulation lasts for seven years, the Rapture will have to happen before then and so it could happen anytime in the next ten years.
Conclusion
As shown above, the date of the rebirth of Israel in 1948 can be calculated based on Ezekiel's vision and the extended period of punishment when the Jews did not fully repent from Babylonian captivity in 536 BC. This is important because not only does it show that God keeps his promises exactly, but that Jesus said the generation that sees the rebirth of Israel will see his second coming - and there are only about 20 or so years before most of this generation will be dead.
So as Christians, since we know that the length of the Tribulation before Jesus returns is approximately 7 years, and the Rapture must occur before that, it shows that we may experience the Rapture very soon! This should motivate us to do the best we can in serving Jesus, and to make sure we love Him with all your heart, as the end is coming fast and we need to be ready to leave at any time.
[1] Grant. R. Jeffrey (1988) Armageddon: Appointment With Destiny. Bantam Books, New York, United States. ISBN 0-553-28537-8.
[2] http://en.wikipedia.org/wiki/Israel#History
[3] http://en.wikipedia.org/wiki/Figs_in_the_Bible