i googled "why is my embroidery floss fuzzy" & unlocked some advanced knowledge (there's alt text if the image is too small to read)
by "right end of the skein," i assume they mean "right" as in "correct," not "right" as in the direction, because that wouldn't make sense since you can hold it in any direction. but the "correct" end on DMC floss is going to be the end closest to the number band.
i'm going to try using this advice to see if it helps me achieve smoother results, because usually it's fine but sometimes my thread is just tore up & my lines end up all fuzzy. (my mink stole piece was like this & it made me so upset because it was a gift!) others in the reddit thread mentioned that this can also be caused by using a needle that's too small or has an eye that's too small for your floss, which causes it to fray when it interacts with the friction of either the cloth or the eye, or both. they also suggest using shorter lengths of floss or using a product like thread magic, though the latter seems like an annoying extra step to me because i thought that's what mercerization was for.