1. "With school turning out more runners, jumpers, racers, tinkerers, grabbers, snatchers, fliers, and swimmers instead of examiners, critics, knower and imaginative creators, the word 'intellectual,' of course, became the swear word it deserved to be. Surely you remember the boy in your own school class who was exceptionally 'bright,'...and wasn't it this bright boy you selected for beating and tortures after hours?"
In this, Beatty is saying that in their society, being smart is not valued as much as being athletic or daring, or doing action-oriented things for the thrill and excitement. He relates it to the smart kid that everyone makes fun of. If you think of your grade, and the super smart kids, don't people make fun of them? Don't they get picked on? They are apart, not part of the "popular" crowds. Being smart simply isn't cool like being a really good football player is. So, intelligence, reading, and book-smarts are looked down on and made fun of in their society. This leads to a generalized "dumbing-down" of their society, and pretty soon no one thinks on their own anymore. I can see that happening in our society too.
2. "We must all be alike. Not everyone born free and equal, as the Constitution says, but everyone made equal."
In this quote, Beatty explains the trend that their society has of trying to force everyone to be the same. In our society, there are many policies and situations where that also happens. If you make a lot of money, you are taxed more than if you don't make very much money--this "forces" the wealthier to have less money so that they are more similar to the poor in wealth. So, our society is alike in that way too, in many ways.
3. "Classics cut to fit fifteen-minute radio shows, then cut again to fill a two-minute book column, winding up at last as a ten-or twelve-line dictionary resumre."
In this excerpt, Beatty is referring to the fact that books were shortened and shortened and shortened until finally no one read the book anymore, they just read a little summary about it in the dictionary, so that they could feel like they read the book without having to go to all of the work of reading it. So, when is the last time that you know of someone who watched the movie version of the book so that they wouldn't have to read the book? It happens all of the time. People don't take the time to read a book these days--they are so long! They take so much time and effort! And why bother when you can watch the movie or read the Cliff's notes?
These are just three things that are true about our world; there are many more, but I hope that gets you started. Good luck!