Is it true that absence of evidence is not evidence of absence? The best answer is: It depends on the case. Let us consider three cases:
I. There is no evidence for the existence of extraterrestrial life, but the absence of evidence is not evidence that there is no extraterrestrial life. This is because we may have reason to believe that it is improbable for life not to exist somewhere other than Earth.
II. You are investigating a murder case and there are only two persons, A and B, who might have motivation for murdering the victim. A and B are the only suspects; you have some (but not overwhelming) evidence that A committed the crime, but you do not have any evidence that B did it. The absence of evidence that B did it is not evidence that he didn't do it --- he is still a suspect. However, if later on there is more and more evidence that it was A who committed the crime, this will allow you to give more weight to the fact that there is no evidence that B did it, and you can accordingly say that the absence of evidence is now evidence that he didn't do it.
III. Suppose Jennifer believes that there are unicorns. You point out that unicorns are mythical creatures and there is no evidence that they exist. She responds by insisting that the absence of evidence for the existence of unicorns is not evidence that they don't exist. Is this a reasonable response? No, for we have very good reasons for believing that there are no unicorns and there is no need for having any evidence for the truth of the belief.
There can be a more systematic treatment of "Absence of evidence is not evidence of absence"; the point I want to make here is just that we should not accept it as a general principle.