It's a rather confusing way of referring to a "biased point of view". Saying that "Person A has privilege" wrt. some issue is a claim that A's overall observations and experiences are unrepresentative, and so she should rely on others' experiences as much as on her own.

That's not quite correct; I think it's best to start with the concept of systematic oppression. Suppose for the sake of argument that some group of people is systematically oppressed, that is, on account of their group identity, the system in which they find themselves denies them access to markets, or subjects them to market power or physical violence, or vilifies them in the public sphere -- you can provide your own examples. The privileged group is just the set complement of the oppressed group. An analogy: systematic oppression is the subject and privilege (in the SJ jargon sense) is the negative space.

The "biased point-of-view" thing follows as a near-corollary because it's human nature to notice one's oppression and to take one's absence-of-oppression for granted as a kind of natural status quo, a background assumption.

Next question: in what way did Aaronson's so-called wealthy white male privilege actually benefit him? To answer this, all we need to do is imagine, say, a similarly terrified poor black trans nerd learning to come out of their shell. Because I've chosen an extreme contrast, it's pretty clear who would have the easier time of it and why. Once you can see it in high contrast, it's pretty easy to relax the contrast and keep track of the relative benefits that privilege conveys.