When people say 'privilege' they can mean many different things. For the rest of the article when I talk about privilege I am using the social justice definition to which privilege means the advantages that non-marginalized groups have.
As a white cisgender heterosexual woman I have quite a few privileges myself, but those privileges are different than if I were to be a white cisgender heterosexual man. Intersectionality goes and looks at these axes of identity and sees how they interact to create discrimination and oppression. If you're interested in learning more about intersectionality I recommend reading Mapping the Margins: Intersectionality, Identity Politics, and Violence Against Women of Color by Kimberlé Crenshaw who originally coined the term.
Understanding your own privileges means also understanding how your own axes of identity intersect and how they interact. Things people can see about you and things you can't hide in your sleeves and pretend they aren't there usually are bigger effectors of privilege. By this I mean; I'm an atheist you can't tell by looking at me that I am an atheist (my shoes definitely don't say atheist), so anyone who might discriminate against non-christians would not be aware unless I said something. However, a woman who was non-gender conforming was denied a tip because of her appearance.
Privilege is just the other side of oppression.
Discrimination: Varying treatment of a group because of their axes of identity.
Oppression: Is systemic discrimination carried out by those in power.
Without understanding what your privilege is or what advantages it gives you it's almost indistinguishable from luck. Thoughts like, "I got this job because I was lucky" may mask realities like, "I got this job, because I had enough class privilege to be able to quit my job and find a better one". Or on a different axes of identity, "I got this job, because I am white and the interviewers subconscious preference for white people helped me".
You may be cringing by now. In fact I'd be really surprised if you aren't cringing. Probably thinking "that's awful, I got where I am because I am awesome" or "I had a hard life too" or if you have already accepting privilege as shaping your position in the world, "I hate that's the way it is". That last thought, that's the one that takes a long time to get to if you have a lot of privilege to grok.
When you have a lot of privilege and someone tells you have beneficial it is to your life it can feel like someone telling you that you got where you are through no act on your own. It can feel like someone saying, "That 'free will' you think you have, yeah... not a thing". But here's the thing, free will versus determinism is a philosophical debate. Privilege vs some mythical idea of a meritocracy is not.
There's a lot of solid evidence that people are treated differently based on the perception of who they are. To see this evidence all I takes is to pay attention to the news, or watch any sort of television, read a book (I highly recommend A People's History of the United States), consume pop culture.
Accepting and processing all of this is hard. Understanding your privilege doesn't mean trivializing your accomplishments. It doesn't mean you're a bad person for having privilege. However, until you try to unpack and understand the privileges you have it's difficult to empathize and understand the struggles of those who do not have the same privilege and empathy is really just the first step to enacting real change.