The double standard they complain of is that Black people are allowed to use it, but white people aren't.
And superficially, you can see why they'd think this was a double standard, and a racist one at that. After all, if the only discriminant on who can do something (whether it be using a word or a water fountain) is the colour of their skin, then gosh darnit that's RACIST! But maybe there's a better way to understand this.
Let's look at pronouns, particularly first and second person. These are words which literally change their meaning, depending on who is speaking. When I use the word "I", it actually denotes a different individual from when you use the word. It's the exact same word, but the meaning is different. And if I type the sentence "I am the author of A Blog Of Tom", the sentence is true when I say it but (probably) false when you utter it, unless either you're me or you happen to write your own blog which just happens to have the same title as this one.
No one has trouble with this concept, once they master English or any of the hundreds of other languages that have relative pronouns, or indeed words like "here" and "there" or "now" and "then" or "tomorrow" and "yesterday". These words mean different things depending on where or when they are used or who is using them.
Well, that's kind of how it is with the N-word. Think of it as a special kind of pronoun. When a Black person uses it, it can have a meaning roughly like "one of us", whereas when a non-Black person uses it, it cannot help but mean "one of them". Of course, unlike basic pronouns, this one has a whole lot of other connotations loaded into it. As "one of us", it is inclusive, hinting at shared understanding and experience; as "one of them" it is inherently exclusive, and implies a sense of disrespect, contempt if not outright hatred.
So the answer to the question about who can use the N-word is really this: anyone can. It's like I taught my son when he was very little about profanity: I don't care what words you use, so long as you use them appropriately and correctly. As a white person, I could use the N-word if I wanted to, if the meaning I was trying to express was "those contemptible people". But I don't feel that way, so it would be a lie for me to use the word. And if I tried to use it to express the sense "one of us", no one would read it that way, because as a white person I don't get to use the pronoun "us" to talk about a group I don't belong to.