Why does the Bitcoin core client use so much CPU time while synching?
I would imagine verifying all of the signatures of the blocks? 


You are verifying every single script, every single signature for continuity. You are making sure nobody is spending coins they're not allowed to. You're making sure all the rules are followed. 


  Essentially doing the same calculation the miners seek to do and just checking their answers?


Yes, in addition to the above. You are verifying every block has a valid Proof Of Work. This takes two SHA256 operations though, versus the billions of billions a miner needs to do.