I'd like to explain to you how you have completely missed the most basic definition of privilege; why you, like so many others, immediately got offended at the idea that you have somehow had it easier than others and leapt to your own defense without trying to listen or understand what someone was truly saying.
Privilege is when you get conscious or unconscious benefits from a demographic trait about yourself that you cannot control. These benefits may be overt (getting paid more as a man) or they may be covert (being able to walk down a street alone at night without fear of violence). I want to stress the last type of privilege. A privilege does not have to be something positive; it can simply be the lack of something negative.
Privilege is not personal. Privilege is institutional and cultural. It is macro. You have privilege because you are part of a group that has privilege. It is not because you are special or different or better in anyway (any more than those without privilege are not special or are worse in any way). This is going to be really hard for you to hear, and this is normally about the time where you're going to start railing and ranting about how you are at Princeton because you worked super hard and are naturally brilliant and wonderful. Maybe you are; and no one wants to take that away from you. Seriously - no one is saying that. But let's step back and remember once again, this isn't a personal conversation.
Checking your privilege doesn't mean anyone is asking you to say "I only have things because I am part of privileged groups". It does mean someone is asking you to say "By position of a characteristic I was born with, I have been helped, or at least not hurt, more than others without this characteristic". It does not mean anyone wants you to apologize for it; it does mean someone is asking for an acknowledgement of the implications of it, either for how it is impacted where you are now, how it might be skewing your perspective or level of knowledge in discussing a subject, or for how the lack of that same privilege may have made things different for someone else.
You may be in Princeton, but it seems like we should probably put this in really simple kindergarten examples for you. In the simplest, crudest metaphor I can think of, let's say you're a fully abled person in a race against a man with only one leg. You train a long time, run really fast, and beat him. No one is saying you shouldn't be proud of working hard or running so fast; all we're really asking for is that you admit that maybe having two legs fucking helped a little bit.
Using this metaphor, let's again break down some other arguments you can't really use. For instance, just because some one-legged people are faster than some two-legged people or manage to race doesn't mean that it is still not, on the whole, easier for two-legged people to walk and run. Again, privilege deals with macro level institutional and cultural ideas, not anecdata. If your grandfather only had one leg, but you had two, you don't get to claim that you do not have two-legged privilege. Having ancestors that endured hardships is important only if either you endure those same hardships or if those past hardships have continued on today in the form of discrimination based on your shared characteristics.
Privilege isn't an on/off switch. You can have one type of privilege, while not having others. You can, for instance, feel like you don't have religious privilege while still being able to admit that you have white privilege and male privilege. If our two legged man is black and our one legged man is white, he's still got white privilege.
Not all privilege is obvious; above, to make it simple, I used a metaphor where you could clearly see how one person had ability privilege (having two legs in a race instead of one). Some other examples are equally obvious: white/POC, male/female, rich/poor. Some other examples are not. Some require you to examine things you too for granted and realize the privilege underneath them. For instance, you admitted you were privileged because you felt like your parents and grandparents made sure you were educated. Looking deeper, I can reasonably assume that there is a huge likelihood that that involves financial privilege; no matter how wonderful a parent, someone who has to work three jobs in order to put food on the table is less likely to have time to spend learning the alphabet with you, or even possibly to be educated themselves. Looking deeply at things you're offering up as the ways in which you are privileged are going to show you that you are in fact privileged in the ways that you just denied. And again - that's okay. I'm genuinely happy you had that. Nothing to apologize for, just something to acknowledge.