How to wire in my replacement fluorescent light fixture?
Most likely the second white and second black wires are the connection to the switch itself.  If so, then what you've got is power line from circuit box to light, switch line from light to switch.  (The alternate method is power line from circuit box to switch, switch line from switch to light - which would have been less confusing for you but he probably was dealing with old work by the sound of it.

How it works in your configuration (if I'm right) - imagine the electrical current like a flow of water.  The power comes in from the breaker box on the black line (aka the hot line).  The power then runs to the white line and down to the wall switch.  When the switch is on it then runs back up the black line to the light.  You connect black to black, and the power flows into the light, then out the white line.  You connect white to white, and the power flows on down back to the breaker box, and out.

So - don't touch the second white and black.  Just wire the light's white to the lonely white and black to black the same way.