I don't like the phrase because it creates a lot of misunderstanding and some people use it maliciously, but I guess that's the case for a lot of things, especially when context is involved. The phrase isn't meant to shame anyone (though it can be used in that way, context is always important) rather, it's kind of like a nudge to make sure you understand your position when making certain statements. For example, a straight male going on about how homosexuals don't really have it bad and just need to suck it up would be speaking from a point of privilege, or rather, a lack of a specific disadvantage or disadvantages faced by specific people. Since they haven't faced these specific disadvantages, they can more easily brush it off or speak down on it because it hasn't affected their life in any way.
It's not about shame, it's simply the acknowledgement that being male/straight or whatever it is, keeps you from having to face certain disadvantages, like how people of color are profiled by police much more often. So in that case, being white means the chances of being profiled by the cops in America are significantly lower than if you had been born with darker skin. Think U.S. post 9-11 and how a lot of people who looked middle eastern were being profiled as terrorists at airports. Not resembling that "mold" of what terrorists are apparently supposed to look like meant you had a specific "privilege" or lack of disadvantage.
Again, this is why that phrase is terrible. It's much more productive to explain why a person's opinion is misguided with a solid counter-argument than yelling CHECK YOUR PRIVILEGE as some sort of end-all rebuttal.
Edit: Also, LOL @ that YouTube video. Jesus, that is one weird, grating sound the person was producing. I'm really curious what the sign said. Wait, just saw a higher quality video. Sign says: :WARNING: Masturbators, Thieves, Liars, Drunkards, Fornicators, and Homosexuals :JUDGMENT DAY: