Once the frame is complete, the spider stretches silk threads across it, just like the spokes of a bicycle wheel.

Literal

Frame [subject-が] become-complete-[when-と], spider [topic-は] exactly bicycle [possessive-の] wheel [possessive-の] spoke [like-のように], to-it silk [possessive-の] thread [object-を] attaches.

A biological description. 出来上がる (できあがる) 'to be completed, to be done' is a compound verb. やっちゅう wait no — や here is the 'spoke' of a wheel, from 矢 ('arrow'), since bicycle spokes look like arrows radiating out. The metaphor 'like bicycle spokes' is the key image. ちょうど ('exactly, just') intensifies the similarity. 糸をかける ('to hang/attach threads') uses かける for setting up tension lines.