To fussy cut means that you're not going to just cut the piece you need from any old place on your fabric; you're going to be fussy about it and frame a certain pattern or design.
I guess I think it's silly because the word "fussy" is silly to me. It reminds me of when Pam on The Office tells Michael he seems fussy:
Pam: Would you like some aspirin? You seem kind of fussy...
Michael: No, I don't want any aspirin! Aspirin's not gonna do a dang thing, Pam. Of course I'm fussy! I'm sitting here with a bloody stump of a foot!
To me, Ayumi at Pink Penguin is a master of fussy cutting. Check out this patchwork wristlet - how the squirrel and ram are cut out so they're perfectly centered, and how it says "Paris Daisy" in the lower left corner. I love how she plans things out like how the rooster is looking longingly at the apple (flower?) next to him.
As for me, I'm usually very efficient and stingy about my fabric use. I haven't been fussy yet. But I'm thinking about it, and in the meantime, I'll share the term with my student, Raz, and maybe he'll get why I think it's silly.