The origin of the theory

In 1827, John Nelson Darby, argued that the Bible taught a pretribulational rapture, and he convinced many other Christians of this view. Darby's leadership and teachings attracted a growing number of followers, and the movement spread to England and other parts of the world.

Many years later, Hal Lindsey, an American author and Christian theologian, made the Rapture theology popular with a book that he published in 1970, titled "The Late Great Planet Earth" that stated:

The Bible teaches that there is coming a time when Jesus Christ will return for His church. But this will not be His final return to earth. This will be a secret, invisible return in the air to meet those who believe in Him. It will be a moment of unbelievable joy and excitement for those who are ready. But for those who are not ready, it will be a moment of terrible shock and sorrow. The Bible calls this event the 'rapture' which will take place at a time unknown to anyone except God the Father."

-- Hal Lindsey, The Late Great Planet Earth, Chapter 12.

In other words, it seems like he thought that Christians would be unexpected "teleported" to a secret place somewhere in the air to meet Jesus, while shock and sorrow will be upon those who were left behind.

In addition, some Pre-tribulationists often interpret biblical passages like 1 Thessalonians 4:13-17 and 1 Corinthians 15:51-52 to suggest two separate comings of Christ. They see the rapture as a secret, pre-tribulation event to gather believers, followed by a later, visible second coming to establish his reign on Earth.

Proof of the theory

The gathering of the elect

One of the verse that is quoted to proof the Rapture is:

And he will send out his angels with a loud trumpet call, and they will gather his elect from the four winds, from one end of heaven to the other.

-- Matthew 24:31 (ESV)

However, according to some sources verse 31 was added to the scripture and is missing from some texts.

Either way, even if the scripture is authentic, you will notice that the angels gather the elect after the Tribulation and the return of the "Son of Man". This disproves the Pre-tribulationists theory that the rapture will happen before the Tribulation.

Like the days of Noah

Heaven and earth will pass away, but my words will not pass away.

But concerning that day and hour no one knows, not even the angels of heaven, nor the Son, but the Father only.

For as were the days of Noah, so will be the coming of the Son of Man. For as in those days before the flood they were eating and drinking, marrying and giving in marriage, until the day when Noah entered the ark, and they were unaware until the flood came and swept them all away, so will be the coming of the Son of Man.

Then two men will be in the field; one will be taken and one left.
Two women will be grinding at the mill; one will be taken and one left.

Therefore, stay awake, for you do not know on what day your Lord is coming. But know this, that if the master of the house had known in what part of the night the thief was coming, he would have stayed awake and would not have let his house be broken into. Therefore you also must be ready, for the Son of Man is coming at an hour you do not expect.

-- Matthew 24:35-44 (ESV)

Why is the "Rapture" like "the days of Noah"?

Not because Noah "escaped the flood", otherwise Jesus would have said so, but he does not. Instead, he repeatedly echoed that the "the coming of the Son of Man" will be unexpected like the flood in the days of Noah.

However, because Jesus repeated says this event will occur unexpectedly in combination with the words "one will be taken and one left", some people belief this means that the elect would unexpectedly be teleported from Earth.

The original Greek word that was translated to the English "taken" is παραλαμβάνεται (paralambanetai), which is a combination of "pará" which means from close or alongside and "ambánō" which means to receive, admit or acknowledge. In other words it could possibly be explained as to be "accepted into position".

For example the same word was used in these scriptures:

The Lord appeared to him in a dream, saying, “Joseph, son of David, do not fear to take Mary as your wife. -- Matthew 1:20 (ESV)"

and

When Joseph woke from sleep, he did as the angel of the Lord commanded him: he took his wife. -- Matthew 1:24 (ESV)

The original Greek word that was translated to the English "left" is ἀφίεται (aphietai) which means to send away or leave alone, i.e. "reject" or "unforgiven".

For example the same word was also used in the scripture:

But if you do not forgive others, then your Father will not forgive your transgressions. -- Matthew 6:15 (NASB 1995)

Jesus more likely meant, that from human perspective two persons would be considered "equal", but one will be accepted like a bride while the other would be rejected.

This theme, of being ready at any time for the Lord's unexpected return to be accepted as stewards of his kingdom, is also confirmed by the 3 parables that immediately follow this passage:

The faithful and wise servant

Who then is the faithful and wise servant, whom his master has set over his household, to give them their food at the proper time?

• Blessed is that servant whom his master will find so doing when he comes. Truly, I say to you, he will set him over all his possessions.
• But if that wicked servant says to himself, ‘My master is delayed,’ and begins to beat his fellow servants and eats and drinks with drunkards, the master of that servant will come on a day when he does not expect him and at an hour he does not know and will cut him in pieces and put him with the hypocrites. In that place there will be weeping and gnashing of teeth.

-- Matthew 24:45-51 (ESV)

The Ten Virgins

Then the kingdom of heaven will be like ten virgins who took their lamps and went to meet the bridegroom. Five of them were foolish, and five were wise. For when the foolish took their lamps, they took no oil with them, but the wise took flasks of oil with their lamps. As the bridegroom was delayed, they all became drowsy and slept.

But at midnight there was a cry, ‘Here is the bridegroom! Come out to meet him.’

Then all those virgins rose and trimmed their lamps.

And the foolish said to the wise, ‘Give us some of your oil, for our lamps are going out.’
But the wise answered, saying, ‘Since there will not be enough for us and for you, go rather to the dealers and buy for yourselves.’

And while they were going to buy, the bridegroom came, and those who were ready went in with him to the marriage feast, and the door was shut.

Afterward the other virgins came also, saying, Lord, lord, open to us.’
But he answered, ‘Truly, I say to you, I do not know you.’

Watch therefore, for you know neither the day nor the hour.

-- Matthew 25:1-13 (ESV)

The Talents (Matthew 25:14-30)

For it will be like a man going on a journey, who called his servants and entrusted to them his property. To one he gave five talents, to another two, to another one, to each according to his ability. Then he went away.

He who had received the five talents went at once and traded with them, and he made five talents more. So also he who had the two talents made two talents more. But he who had received the one talent went and dug in the ground and hid his master’s money.

Now after a long time the master of those servants came and settled accounts with them.

And he who had received the five talents came forward, bringing five talents more, saying, ‘Master, you delivered to me five talents; here, I have made five talents more.’
His master said to him, ‘Well done, good and faithful servant. You have been faithful over a little; I will set you over much. Enter into the joy of your master.’

And he also who had the two talents came forward, saying, ‘Master, you delivered to me two talents; here, I have made two talents more.’
His master said to him, ‘Well done, good and faithful servant. You have been faithful over a little; I will set you over much. Enter into the joy of your master.’

He also who had received the one talent came forward, saying, ‘Master, I knew you to be a hard man, reaping where you did not sow, and gathering where you scattered no seed, so I was afraid, and I went and hid your talent in the ground. Here, you have what is yours.’
But his master answered him, ‘You wicked and slothful servant! You knew that I reap where I have not sown and gather where I scattered no seed? Then you ought to have invested my money with the bankers, and at my coming I should have received what was my own with interest. So take the talent from him and give it to him who has the ten talents. For to everyone who has will more be given, and he will have an abundance. But from the one who has not, even what he has will be taken away. And cast the worthless servant into the outer darkness. In that place there will be weeping and gnashing of teeth.’

-- Matthew 25:14-30 (ESV)