Free kick after a fair catch?

[stegenref]

OK, so I was studying last night and I came across this rule about getting a free kick after a fair catch (or an awarded fair catch). I've read it several times, but I can't ever wrap my mind around what they're trying to say. Can somone explain it to me and/or give me an example of when this has happened to them? Or why any team would want to do that?

I think there was a case where R received a fair catch at the 50 yard line, ran a play (incomplete pass) and there was a fifteen yard penalty (roughing the passer maybe?), so on the next play the captain of A (previously R) requested a free kick since they were now on B's 35 yard line. How does that work?