Cobwebs
"never been employed"
Dickinson is playing with the word 'employed'—spiders aren't hired as artists, but they do work constantly. The joke sets up her defense of unrecognized labor.
"broom and Bridget"
Bridget was the stereotypical name for Irish housemaids in 19th-century America. The spider's 'certification' comes from being constantly swept away—proof of productivity through destruction.
"I take thee by the hand"
This gesture of solidarity is both absurd (you can't shake hands with a spider) and serious. Dickinson identifies with another small, industrious creature whose work goes unvalued.