It's literary English, not ordinary everyday English. It is a metaphor.
Each "dependency" is like a connecting thread.
For example, I bring my wife coffee every morning. She depends on me for her morning coffee. This is a connecting thread. Of course she knows how to make coffee herself, but she is used to having me make it.
A lot of these connecting threads can become like a net.
"Wrapped in a tightly woven net of dependency" sounds like a bad thing. There is too much dependency. It has become a kind of trap, like a fishing net.