That's not what they are saying. Imagine that you have 24 students. How many ways can you pair such students? 276 ways. Not quite 365 but kinda sorta close. So they are saying that even with 24 students, you have 276 chances that there is going to be a pairing that has the same birthday. And the entire EVENT is that if you go through the entire class you'll find a pair of birthdays, not if you take two people from each class and determine if they have the same birthday. Does that make sense?
For the approximation, you really aren't meant to "understand" anything except how to type in and calculate that formula. They throw a few terms out at you that probably aren't going to mean anything to someone just in pre-calculus. All you need to know is how to calculate the size of your group to get the rough approximation that you want. At the end of their section they used .5 and found that with 365 days, you would need 22 students to get a roughly .5 chance that at least two students (one pair) have the same birthday.
How are you trying to formulate your report? Out of that entire section ther are about two parts that are probably relevant - you'd probably have to explain why some of this stuff works, which would have you going into topics you may or may not have covered in HS.