I am confused about the notion of the significance level ($\alpha$) in hypothesis testing. There seems to be two (at least) ways of looking at the significance level, and I am unable to reconcile them, that is, I don't understand how they are equivalent. From this article, the author first states:
The significance level, also denoted as alpha or α, is the probability of rejecting the null hypothesis when it is true.
Then he shows this normal distribution and goes on:
The critical region [shown in red on the provided image] defines how far away our sample statistic must be from the null hypothesis value before we can say it is unusual enough to reject the null hypothesis.
Why is the concept of choosing a threshold for the hypothesis test related to the concept of having a false positive?