Imagine, for a moment, living in a society in which toddlers routinely stab infants with knives, and adults routinely smash toddlers with rocks. You, naturally, are horrified by this state of affairs, and are thus flabbergasted when everybody suddenly notices that toddlers stabbing infants with knives is a bad thing - and pushes for a solution of increasing the rate at which toddlers are smashed with rocks.
You, having come from moral sensibilities that arose in an entirely different place in time in which stabbing infants and smashing toddlers are both considered horrifying, are naturally going to feel pretty conflicted about this. Yeah, it's great that they've decided that it's a bad thing that toddlers stab infants with knives, but also their solution actually kind of makes everything worse. They've progressed to a morality more similar to your own, but the increased similarity has, in a sense, made everything worse.
Your friend in this society, whose views you find abhorrent, is puzzled by your reaction. Why are you so upset? Sure, you didn't get everything you wanted, but society has progressed, hasn't it? Shouldn't you support this new initiative, which is, after all, a step in the right direction, the direction you want everything to progress in? Why are you so insistent that knives be taken away from toddlers, as access to knives is part of every human's natural rights, when there's a much more moral solution that leaves everybody better off and doesn't harm anyone's rights, which is to say, smashing the toddler's heads in with rocks?
Setting aside how contrived this scenario is, I think it basically describes a routine state of affairs in the world: From any given moral perspective, progress in one moral dimension will basically always amount to regression in another, even if it's only the inverse moral position. We frequently find ourselves biting repugnant bullets, and it's difficult not to be at least a little bit cynical about the state of affairs.
It's hard not to get annoyed at those who never have to make these trade-offs, particularly in situations where they get upset about having to make a trade-off, or acknowledge a moral perspective they don't really care about. Because the privilege referenced in the title of the post relates to the idea that a majoritarian moral perspective may find itself making no moral trade-offs; that is, a person who agrees with the majority moral perspective on every single issue may find society to be a wonderful place with constant moral progress in the right direction, and more, may find anybody who feels anything less than enthusiasm about the progress of society to be, well, evil.
Where the rest of us find ourselves dealing with people who act like there aren't any trade-offs whatsoever, like their solutions to problems, which happen to maximize all of their values as much as possible, are completely perfect, and we are absurd or evil for questioning the value of those solutions.
In our toddler-smashing society, the radio is full of songs about smashing toddlers' heads in with rocks. There used to also be songs about toddlers stabbing infants with knives, but as sensibilities have changed, now those songs are getting censored, and removed from the radio. So now all the songs are about smashing toddlers' heads in with rocks. Since this society has only two hats, and they've hung one of the hats on the rack, that's all the songs they have; you've lived in this society for a while, and, while you don't like the content of the songs, they are all the music that exists, and you've come to be able to appreciate the artistic qualities of the songs in spite of their lyrical content. You find yourself objecting to the censorship; they're censoring half the good songs that exist.
Your friend is puzzled why you're annoyed about the censorship - you're against stabbing infants with knives, after all, why are you complaining about it? All decent people who hear those horrible lyrics are naturally offended, so why should those songs get any airtime? Are you secretly for stabbing infants with knives? Censoring the old infant-stabbing songs is pretty much the only decent thing to do.
And it's difficult to express the problem you have with these arguments, because they're reasonable, in isolation. The problem is that -all- music is offensive to you, but only -some- music is offensive to them; so you've been forced to get used to hearing offensive views, and so offensiveness has become somewhat tolerable. They, however, are not used to hearing offensive music - the experience is raw and painful to them.
They have a kind of moral privilege, with respect to you; they never have to experience living in a society full of moral evils, whereas that is your default experience. Everything exists to cater to them - and they get rid of anything that doesn't. Their newfound morality, even though it is agreeable in an of itself, has come with some profound losses, losses which are justified entirely on the basis of their own unwillingness to experience anything which they find disagreeable.
Now, we could construct a utilitarian argument for censoring the music in our toddler-smashing society; you're literally the only one who is upset by the loss of the music, the only one who had to put up with disagreeable moral notions in the first place. But that's somewhat beside the point.
I don't have any particular recommendations here; mostly my purpose is to gesture at, and maybe help other people notice, something which I think encapsulates a common experience, because I think it may be a useful concept to some people.