How to use the 1/e law of best choice?

Suppose that all applicants have independently of each other the same arrival time density $f$ on $[0,T]$ and let $F$ denote the corresponding arrival time distribution function $$ F(t) = \int_0^t f(s) \, ds, \, 0 \leq t \leq T. $$

The author is stretching the analogy somewhat to give intuition for the integral of the probability density function. In this model, you might tell the applicants to email their resumes from midnight this morning to midnight tonight. Suppose the resumes arrive randomly according to the probability density function $f(s)$. Let $\tau$ be the time such that $F(\tau) = \frac{1}{e}$. In other words, a $\frac{1}{e}$ fraction of the area under the probability density function has been witnessed, which should roughly correspond to having seen a $\frac{1}{e}$ fraction of the applicants. (In practice, you could be so unlucky that all applicants randomly choose to arrive just before midnight tonight, but this is highly unlikely.) The strategy recommends that you dismiss all applicants who emailed before time $\tau$ and hire the first applicant after that who is better than all the previously dismissed ones.

In short, your goal is to reject the first $\frac{1}{e}$ fraction of applicants and hire the next applicant who is better than all the previously dismissed ones. If you happen to know there will be exactly $N$ applicants, then you can dismiss the first $\frac{N}{e}$ of them. If you do not know how many applicants will apply, then you must make some judgement about the distribution of the application times and choose the time when you expect a $\frac{1}{e}$ fraction have already applied.