Privilege is a lens through which you look at the world and analyze
things. It's more or less an analytical tool developed as a response to various flavors of social darwinism.
The idea behind privilege theory is to say that, when you compare people, you don't say "the more successful one is more successful because they are better", but instead "there are a number of factors, many of which are difficult to quantify, that contribute to the varying levels of success between two people (or especially groups)".
It's not one dimensional, and there's no objective way to measure or compare things as generic as "success". Privilege theory being explicit about this makes it superior to many of the alternatives.
"Do what" is such a broad question that I can't really answer it. Are you an academic, or a company or a government or something else? The answer depends on that. It's a malformed question, like asking "What do you do with Kantianism?" or something. The best answer I can give is "use it as an analytical lens to guide your decision making", which is a bad answer precisely because its a bad question.
A better one might be "What value does analyzing situations/systems via the lens that privilege provides give?" Which is incidentally also a pretty good question to ask about Kantianism.
The answer to that question also depends on exactly who you are and what you're doing, but at least it makes sense. For example, for me personally, I think that looking at things via a lens of privilege helps me to be a more empathetic person, on the whole.
For an academic, it might help you decide whether or not you are asking the right questions in an experiment.
As a very straightforward historical example, for many years in the US, there was explicit bias against African Americans in segregated schools. This was justified in many cases, through a social darwinistic lens: black people are less intelligent, the proof is that they perform worse, and so we should devote fewer resources to their education.
Looking at the same data through a lens of privilege though, one might instead conclude that segregated schools performed worse because they had fewer resources. In other words, the same correlation is explained in two vastly different ways.
I use that example specifically because today the same thing still happens, as a result of most school systems getting funding based on property taxes, and historical reasons so you still have a world where, in practice, predominantly black schools have much less funding. Ostensibly, this is "fair", since you use the same rule to fund everyone, but clearly it isn't actually. Privilege is one way of accounting for how things that are ostensibly fair may not be in practice.
As an aside, in a much more mathematical sense, I think that privilege theory is all about picking the right bayesian priors when doing an analysis.