Then Underwood started the car, closed the garage door, and walked away.
I thought walk away usually meant a person move away using his/her feet
, but here doesn't.
You are incorrect. Actually, it does mean this.
Underwood used the car to walk away, why?
Underwood did not use the car to walk away, he used his feet.
Can walk away be used in such situation where a person uses the bicycle, car, or something alike?
That would be odd.
Now let's look at the scene in which this "walking away" happens. It has to do with Underwood murdering an underling named Peter Russo by leaving the drunk Russo in his car with the engine running, in a closed garage. The fumes build up in the garage, come into the car and Russo dies of poisoning.
The excerpt from the audio, which is faithfully represented in the link to the transcript which you provide is thus:
SPACEY: (As Frank Underwood) Whatever it is you have to face tomorrow, you don't have to face it now. Right now, it's just you and me.
COREY STOLL: (As Peter Russo) I failed myself. I failed my family. I'm so tired.
SPACEY: (As Frank Underwood) You just close your eyes.
DEGGANS: Then Underwood started the car, closed the garage door, and walked away. That murder transformed Underwood from an antihero into a straight-up villain.
(The summary of Season 1, Episode 11 on wikipedia confirms this for us:
Frank picks [a drunk Russo] up from jail and, recognizing him to be too much of a liability, proceeds to kill Russo through carbon monoxide poisoning, making it look like a suicide.)
Russo is left to die in the car. Underwood walks away on his feet.