Take care of the injured crewman in the transporter room first.
When you healed him, he should have pointed out a panel for you. There's a device in there (the "transmogrifier") that should be able to fix the transporter.
Have Spock try to do the repairs.
First use Spock on the console and then use the transmogrifier on it.
You'll learn that you need a comb bit to use the transmogrifier.