The weaving spiders are a reference to Arachne. See Arachne vs Athena in the myths.
The symbol of Athena is the Owl. When old orders of people had an owl and phrases equating to "weavers not allowed here" - it was an expression that could have several meanings.
In a literal sense, it could be a claim that "the little people do not belong here." Arachne, a commoner and human, challenged a goddess to an art/craft (depending on whose story you believe, either Arachne or Athena won - Arachne's conversion into a spider was either the spiteful act of a goddess shamed or a punishment dealt to an arrogant mortal).
Taking a more philosophical approach, it is more commonly held to be an expression that your daily affairs are of no concern to the group. No one cares about your day job. No one cares about how much you hate your boss, how good you banged your wife, etc - the issues the group intends to discuss are 'the realm of gods' and should not be contaminated by the concerns of weavers.
It's a sentiment with a point, but can obviously feed into certain mentalities prone to domineering schemes.