It's quite simple really. Jesus was here talking about the unbelieving family members opposing the Christian in the household.
The ironic view of the WT society is that Jesus was talking about the Christian being the one who cuts himself off from the rest of the family, when he is clearly talking about the other way around. He is talking about a Christians enemy could be within his own house, it is therefore the unbeliever who may cause this division, whereas the WT says it should be the JW who cuts himself off from family.
If the unbeliever opposes the Christian in the household, v 37 then goes on to say that the Christian should not have greater affection for father or mother than for him, what he doesn't say is that they should be cut off and never spoken to, never invite to family gatherings, etc, etc.
34 Do not think I came to put peace upon the earth; I came to put, not peace, but a sword. 35 For I came to cause division, with a man against his father, and a daughter against her mother, and a young wife against her mother-in-law. 36 Indeed, a man’s enemies will be persons of his own household. 37 He that has greater affection for father or mother than for me is not worthy of me; and he that has greater affection for son or daughter than for me is not worthy of me.