Is it mostly because of the antisocial tone of this comment, or is it Nietzsche himself that caused the downvotes?

I didn't vote on the grandparent comment, but I probably would've downvoted it if it wasn't already far into the negatives, because it gives advice that would be terrible and destructive. While there may be an interpretation of

Practice exercising your will, over yourself at first, then over your pets, your friends, family, co-workers and so on into the larger world

which is reasonable, the naive default interpretation of this, and the one that might actually be followed, caches out to "attempt dominance moves over your friends, family and co-workers". Doing that without sufficient skill (which the person this advice was directed at almost certainly lacks) will simply alienate them. And actively seeking "challenge, danger, conflict and conquest" - well, two of those (challenge and conquest) are good things to seek, but the other two (danger and conflict) are stupid and destructive.

Oh, and also, using a throwaway account with "Sith Master" in the name suggests to me that the advice might have been bad in a way that wasn't accidental.