I have Cable box "Explorer 8300HD" and would like to connect to a Sony DVD HR-XVC27U to record shows from the cable box to DVD?

You need to hook one of the outputs from the 8300 into a corresponding input on the recorder. Which one you choose depends on what they have and how you want to hook to your TV.

I think you start by buying the right cables that your boxes requires to connect them together. These links should get you started.

Connecting and Setting up your DVR / DVD Recorder (http://support.toshiba.com/support/viewContentDetail?contentId=3410324)

How to Connect a DVD Player, VCR, and Digital Cable Box: 14 Steps (http://www.wikihow.com/Connect-a-DVD-Player%2C-VCR%2C-and-Digital-Cable-Box)