It means that the world is turned upside down. The wren is a somewhat timid bird, but the eagle is a bird of prey. Shakespeare has turned that around and amplified it. Not only is the wren a bird of prey, but it even goes in search of prey where eagles are too timid to even perch.
In Measure For Measure, Shakespeare says that 'liberty plucks justice by the nose', with a similar meaning.
In modern English, when we want to say that things are the wrong way round, or that the world has gone mad, we often say: "The lunatics are running the asylum."