Here “subtle web” means a web that is not easily visible. When the spider was sure to have flattered the Fly to such an extent that she would surely return to him, he went into his parlour and wove a subtle web as a trap for the fly and also set his table to dine on the fly.
 
When an evil-minded man builds a trap to befool someone, he certainly tries to make it as invisible and hidden as possible and pretends that he is rather a well-wisher. Here that is the case with the spider making the web as a trap for the fly.