I would agree that this is somewhat poorly worded, which makes me hesitant to write this as an answer, but I will, and others can chime in if they think this should be a comment.
Claiming "rights" to something is predicated on the idea that there is something inherent about access to that thing, and from that it follows that something should guarantee access to that thing, whether we are talking about a right to be born, bear arms, vote, speak freely, peaceably assemble, breath clean air, live indoors, marry who we want, not starve to death, access medical treatment, die on our own terms, have an abortion, bomb other countries own a car/house/tv, etcetra. Rights can't be claimed without appeal to something external; even if that thing is decency or shared morality it comes from something external to- and therefore above- the individual, which is the unit to which things will ultimately reduce.
"Fairness," too, appeals to morality, to a presumed shared sense of right and wrong. In an anarchist context this is yet another attempt towards a proscriptive solution. You can't presume to know the particulars of my life, nor could I yours, so I can't say what's right or what's wrong for you.
When one supposes they have "rights" which are then denied them (which is certainly not "fair"), it follows that one has suffered and "injustice." Injustice presumes justice. Justice, like rights, assumes external powers which arbitrate what is and is not just. This is one reason North American anarchists, of late, appeal not for justice or its counterpart accountability, but in vengeance - taking action of ones own initiative.
I am not addressing the idea of "level ground" at risk of veering into an over-long discussion of the failures of reformist identity politics, except to say that we don't all start on level ground, not even in any anarchist future I would envision. It would be nice if we did or could, but again, forcing that would be decidedly non-anarchist.
After all that wrecking-ball, the counterpoints: Yes, unrestrained pursuit of what I want and desire will put me into conflict with others pursuit of what they want; at some point one or the other of us loses or acquiesces leading (quite likely) to oppression. That oppression (or more accurately, domination) puts me in constant conflict with those I'm oppressing, what follows is theft, slave revolts, the treatment given Mussolini. This is the Hobbesian view of it all, and it is not entirely wrong.
On the other hand, if I recognize that my desires will be more hampered by constantly fearing a slave revolt and being strung up, and if I have some sense of solidarity (seeing my struggle/suffering reflected in that of others) especially if I have any sense of empathy (ie - I'm not a sociopath), then I might conclude that many of my strongest and most base desires are best met if I don't impinge on other people's needs and desires to the best I can help it. The reason we do unto others as we'd have them do to us is that if we don't, they are more likely to do unto us as we did unto them. Also, I'm not convinced that prioritizing my own whims and wants precludes me from having the whim or wanting to just be a nice person. I do lots of things because it gives me more pleasure to give other people pleasure, or comfort, or ease.