No idea really. :) Maybe it's from the softdrink/icecream dripping down the sides of the glass... like a spider sitting on top. I've never thought there was a strong connection to the arachnid, in fact. That would really put a child off one! :D I found a page which "explains" the meaning, but as one commenter wrote: poppycock.

I’ve never heard the term “spider” used in the US to refer to a soft drink with ice cream. In my part of the South we call them “floats.” The most popular one is a root beer float, ice cream floating in root beer. In my parents’ generation there was a term “purple cow” used to refer to a coke float. It may have been a purely local term. The word “spider” is used in many contexts here but I’ve never heard it used to refer to something we eat or drink, although in the warm, humid climate of Louisiana we have all sorts of insects, arachnids and their arthropod cousins.