To have completely misunderstood something. The meaning is identical to the more common idiomatic phrase 'The wrong end of the stick.'
The origin of this phrase may be an answer from the Australian edition of The Weakest Link which featured in the satirical magazine Private Eye.
Chair (Cornelia Frances) - "When you have misunderstood something you can be said to have gotten the wrong end of the ... ?"
Contestant - "Dog?"