"[Such-and-such group] has suffered, You People haven't taken adequate notice of it, and I deserve center stage and a License to Overact for pointing it out."
This maneuver consists of popping up in the middle of a conversation and invoking the sorrows and abuses suffered by some group as an excuse to grab center stage, deliver loud superficial condemnatory lectures to those present, and in general demand huge amounts of unwarranted attention.
If you're talking about the complicated history of tri-racial isolate communities in the Appalachian uplands, this is the guy who turns up and demands to know how you dare discuss such a subject without mentioning that Europeans stole the Americas by committing genocide upon its indigenous peoples.
If a secular Western site has run a piece about burqinis, and in the comment thread some Muslim women are having a genuinely interesting discussion about how much they like them, the moral claimant guy is the one who comes in and immediately starts hectoring everyone (in fairly crude terms) about the iniquities of Islam, and how it oppresses women.
The most striking characteristic of people who pull this one is that they have a very superficial understanding of the subject under discussion, and aren't interested in learning more. They have their simple talking points -- Europeans oppressed First Nations, Islam oppresses women, etc. -- and react to people holding more complicated conversations by punishing them for not discussing in on the level at which their talking point would be appropriate. If the other people in the thread then discuss the point they've raised in more detail, they vanish.
Another characteristic is that they take no notice of members of the supposedly oppressed class they've invoked who are already part of the conversation. They'll yell at members of the First Nations about European oppression in the Americas, and yell at Muslim women about Islam's oppression of women.
Like so much other trollish behavior, this maneuver has nothing to do with its ostensible content. It's all about grabbing center stage, posing as the guy who holds the moral high ground, and yelling at everyone there for failing to make a place for him to use his one little talking point.
See also: Asserting an Obligation to Debate.
Note: This maneuver is structurally similar to the Spanish Prisoner con game, The Lurkers Support Me in E-Mail (and related ploys), and the transactional analysis Rescue Triangle phase in which the supposed Rescuer takes the supposed Victimizer to task on behalf of the supposed Victim. In all of them, an interaction that's taking place between two entities is informed by a hypothetical third entity on whose behalf demands are made.
This third entity/Victim need not be present, and may not actually exist. The Rescuer need have no demonstrable connection with them, nor know much about them, as long as he or she can make demands on their behalf. In its more toxic forms, anyone who questions the Rescuer's right to do this may be accused of being an oppressor. If this adversary is a member of the oppressed class previously invoked, they may be accused of false consciousness, colluding in their own oppression, et cetera.
In terms of everyday use, Collecting on Moral Debts is most closely related to My Issue Is Important; Yours Is Trivial.