Hello,
The following is the "of" definition of Longman Dictionary.
In the mind, here the "of" shows that the thing is as a result of "spiders" and "them".
The Longman dictionary regards the "spiders" and "them" as the end of the thing.
I think it to be wrong in logic.
Ask your witness.
12used after some adjectives that describe feelings, to show who or what the feeling is directed towards:

He's always been frightened of spiders.

Most children want their parents to feel proud of them.