Spilling water on tablecloth, by right of sympathetic magic, indicates that rain is on the way. If you drop a glass and it doesn't break, this is proof that you have friends who would go through fire for you.

What does "by right of sympathetic magic" mean here?

Does "sympathetic" mean "positive" here? "Positive magic".
Google 'sympathetic magic' and you will find a more detailed explanation, but basically it is using like to magically cause like (here, the water spilt brings rain water). 'By right of' here is intended to mean 'by way of' or 'by the law of'.
Thanks for this answer!