Since you've stated that you're doing your best to understand, I would like to assist in that. I understand that these concepts are a little slippery at first, took me FOREVER to get them. So, I'd like to hop in and make a couple points. I hope this will help explain things a little better.
No one's saying that everyone who has privilege, has an easy life, or has everything handed to them on a silver platter. We all suffer and have unique problems brought on or exacerbated by the circumstances of our lives.
Anyone who belongs to a minority group tends to suffer at the hands of a majority group (even if some of the individual members of that majority group don't hold prejudices or malice towards related minority groups). The majority group in that case has a certain privilege related to that circumstance in their life where they belong to the majority group. The majority group benefits just by being a part of the majority group.
Understanding how every one's unique circumstances have an effect on their lives and views is part of the intersectionality of it all. Intersectionality is the acknowledgement that we all have our problems and we all work together to help lessen the oppression of others, especially in spaces where we are a part of a majority group.
Example, since we're both atheists we'll start with something easy, like religious privilege. Here in the U.S. it is totally a thing, especially if one is a part of a mainstream Christian religion. If one is a part of this majority group one benefits by feeling a part of the majority, more trust from fellow religious people, more likely to get support from other religious people and are assumed to have somewhat good character just for being religious. I'm sure you could help me with a litany of disadvantages that atheists face in mainstream society that are related to religious privilege, you've already mentioned that being openly atheist can hurt one's chances of being elected to office, we are assumed to be immoral and have bad character. Most religious people are completely unaware that they benefit from this system, if asked would likely say that they (as an individual) do not oppress people, most would obstinately deny that non-religious people suffer for not being religious, that is part of the religious privilege in play. The thing is that it's part of an overall cultural system and not due to individual action.
We can take this mutual understanding and use it to extend the idea to other marginalized minority groups, as long as we keep in mind that the problems associated will vary along with the prejudices and stereotypes associated with each group.
The best way to gain understanding of the problems faced by any group is to listen to people. There's no need to argue, no need to explain, just listen and keep things in mind. When one is part of a majority group and is an ally to a related minority group, one can speak out against stereotypes, slurs and prejudices that come up against that minority group, which bit by bit will help to change minds and alleviate oppression.
When I'm with a trans* person, I listen when they talk about what their lives are like. I don't know, so I listen to gain an understanding. You'd be surprised just how much hate trans* people get, there's a whole lot of mainstream allowance for cruelty and shunning merely based on one's trans status. When I'm with cis people (cis just means your gender matches your biological sex) and something ignorantly negative about trans* people comes up, I speak out and stand up as part of a majority group to help fight negative stereotypes, prejudice and violence against a marginalized group to which I am an ally.
When I'm with a non-white person, I just listen. I don't try to lecture about "reverse-racism" or relate about some obscure negative experience with some other random non-white person. I just listen. When I'm hanging out with white people, I speak out against casual racism (shitty remarks/jokes) so that people understand that these sorts of attitudes and behaviors are unacceptable.
Being queer, I face a lot of trepidation meeting new people. I never know who's going to be friendly or who's a homophobic bigot. There's always that tense moment where I'm just waiting for the hammer to fall, and that's just in light social interactions, it doesn't count the time I was assaulted for my sexual orientation. Now, when I talk about this stuff, I don't need any one explaining things away, I don't want to hear about how it's not that bad and its just like (insert random decently unrelated circumstance here). I'm just relating and would like to be heard without some straight person trying to tell me how it's just a little thing, because it isn't, when it's every fucking time.
These ideas are excellent tools, with which we may help each other work towards a better, more understanding community and a more equal world.
