I'm not sure if you're looking for an ELI5 of the question or the answer. I'll explain the answer since others already explained the question.

You'll only have the time at which your friend leaves (in seconds since the start of the exam). Knowing only this, whatever you two scheme, you'll have to write down your answers (from a scheme you both agreed on beforehand like "leaving after 324 seconds means answer true for the first 7 questions and false for the rest", "leaving after 325 seconds means answer false everywhere", ...) once your friend leaves. So really, you're just a transcriber. During the exam, its your friend who decide what you're going to write down by picking the right second to leave at.

Of course, they'll pick the second which makes you write as many correct answers as possible. (And of course, you'll want to pick the best scheme to help with that. But let's see what you can't do, no matter the scheme, first.)

Why can't you just always get all answers right? Because there are too many possible set of answers (2^23 for 23 questions) and your friend can only choose from the 90 * 60 = 5400 you two agreed on before the start of the exam.

In fact, you can't even be sure of getting 22 questions right. For each second (from 0 to 5399), there are only 23 set of true answers to the exam (among 2^23) for which leaving at that second gets you exactly one answer wrong (namely, one of getting each of the questions wrong). So 5400 + 5400 * 23 is an upper on the number of sets of true answers for which you'll get at most one answer wrong (using whatever scheme). Since 5400 + 5400 * 23 is still (much) less than 2^23, there will be many set of true answers to the exam where your friend doesn't have a choice but to make you write down many wrong answers.

The same goes for getting 21 answers right. For each second, there are 23 * 22 / 2 set of true answers for which leaving the exam at that time gets you exactly two questions wrong. (Pick the first question to get wrong, pick the second question to get wrong, divide by 2 because there are two ways to pick these two questions.)

5400 + 5400 * 23 + (5400 * 23 * 22 / 2)

is still two small. So there are bound to be sets of true answers where you'll get more questions wrong.

It turns out that getting 20 answers right is possible. Now you have to actually pick a scheme. We can't just make the same calculation and add

5400 * 23 * 22 * 21 / (2 * 3)

(which is now bigger than 2^23) because that's only an upper bound. Each second from 0 to 2399 "covers"

1 + 23 + 23 * 22 / 2 + 23 * 22 * 21 / (2 * 3)

set of true answers (so if those are the true answers to the test, your scheme works). But if for some set of true answers is covered by more than one of the seconds (it happens when there are two different times at which your friend can leave and give you at most 3 answers wrong), it means that some other set of true answers might not be uncovered.

In your scheme, the only thing that matters is which 5400 set of answers your friend can make you write (swapping which answer goes with which second doesn't change anything).

As it happens, people have made tables of the minimum and maximum number of seconds needed to cover all 2^23 possibilities for getting at most 3 answers wrong. And its 4096 seconds which is less than 5400. In fact, the table contain ranges for different total number of questions and different number of questions that you can get wrong.

(I think there's something more interesting about this particular scheme that gets you 4096 but I haven't read all the other comments yet and haven't thought about this questions more since.)