In Matthew 23, leading up to the Olivet Discourse, Christ condemns the scribes and Pharisees of that current generation with a series of rebukes beginning with "Woe to you". He condemns them for the fact that they will beat and murder his disciples (which as we know from the book of Acts, did indeed happen). Jesus tells them the blood of the righteous will be upon them. And when will this happen? Jesus answers clearly: I tell you the truth, all this will come upon this generation (v. 36).
In Matthew 24, Jesus tells his disciples that the temple will be destroyed and describes the great tribulation. In v. 34, He says: I tell you the truth, this generation will certainly not pass away until all these things have happened. He began the sentence with "I tell you the truth" or "Verily". This was used for special emphasis on what followed, as if to say: pay careful attention to what I am about to say.
Why preterism? Because 37 years later, what that generation had not passed away, Jerusalem and the temple were destroyed. Why preterism? Because it fits.
Dispensationalists have two interpretations of "this generation". One is that it means race, meaning the Jews. However, this makes little sense. The tribulation is about the Jews. If generation means the Jews, the Christ is saying, in effect, The Jews will not pass away until these things happen to the Jews. Hardly seems worth the effort.
The other interpretation is that since we are talking about the tribulation, which occurs sometime in the future, generation refers to the generation that will be around during the tribulation. That has a similar problem, for Jesus is merely saying: Some you around during the tribulation will not pass away during the tribulation.
It is not just the Olivet discourse that has an "about to happen" tone. So does the book of Revelation. Recall how it opens:
The revelation of Jesus Christ, which God gave him to show his servants what must soon take place. (Rev 1:1a)
Blessed is the one who reads the words of this prophecy, and blessed are those who hear it and take to heart what is written in it, because the time is near. (Rev. 1:3)
The proclamation of the near term fulfillment is repeated at the end of the book:
The angel said to me, "These words are trustworthy and true. The Lord, the God of the spirits of the prophets, sent his angel to show his servants the things that must soon take place." (Rev. 22:6)
Then he told me, "Do not seal up the words of the prophecy of this book, because the time is near. (Rev. 22:10)
There are other verses in Revelation (1:19, 2:16, 3:10-11, 22:7, 22:12, 22:10) which also indicate that John expected a near term fulfillment. Preterists ask this question: If you think passages such as these do not imply imminence, then what could John have written differently if that was his intent?
We could go on and on about passages that speak about something happening soon. Let's just take one more, also from the book of Matthew:
I tell you the truth, some who are standing here will not taste death before they see the Son of Man coming in his kingdom. (Matt. 16:28)
This passage has various difficult interpretations, but not for preterists. Preterists believe that Christ did return in wrath and judgment less than forty years after stating this prophesy, when some of those standing there were still alive. One traditional view of this passage is that it refers to what follows immediately in the text, in Matthew 17, the transfiguration. That is a troubled explanation, given that the transfiguration occurs six days after Christ spoke these words. It means that Christ was really saying: some of you will not be dead six days from now.