DVDdoug:
Some people like to use an external voltage regulator, but the Arduino has an on-board regulator and it's rated up to 20V through the barrel jack or Vin. (I have an Arduino in a car powered by '12V' through the barrel jack.)
The onboard Arduino regulator is unsuitable for the automotive electrical system. That 20V rating is not actually good for 20V. Even 12V without all the nasty surprises can cook the Arduino's regulator under the wrong conditions.
This project is looking to supply many LEDs from that 5V supply. That's exactly the wrong conditions which will cook the Arduino's regulator.
The easiest way to get an automotive-rated 5V power supply is to use a phone charger adapter. $5-$15 at your local discount store will get you something good.
The brake lights connection is much simpler than a power supply. Two resistors configured as a voltage divider will work quite well. A 5V TVS diode or BAT16S shottkey pair after the resistors will protect the Arduino input from problems.