If only an external force can change the velocity of a body, how can
the internal force of the brakes acting upon the wheels bring a car to
rest?

edurbs
          A am always happy to see people who actually think about it,
instead of just memorizing it.

 Road applies external force to the car tires. Easy way to see that
is to imagine a driver, who slammed the brakes and is in a skid.
Friction between road and tires stops car quickly. Out of the skid,
in normal breaking, it is same force, but smaller. 
Again, we brake lightly on slippery road, so that this force 
does not exceed the static friction (which would cause skid).
http://www.20sim.com/webhelp4/library/iconic_diagrams/Mechanical/Friction/StaticDynamicPhenomena.htm

Hedgie

Oh, I just now noticed the comment. It is correct, except of course
the last statement. Brake, the friction inside the disk brakes

http://auto.howstuffworks.com/disc-brake.htm
has important effect, as any driver who got his brakes wet knows.

Commenter probably meant that this force is not stopping the car.
 It just (partly) prevents the wheels from rolling 
(which increases the friction between road and tires)
and that is useful when one wants to stop.

Thank you, I didn't even think about the road/

Very simply - the force of the brakes is not being applied internally
but externally - TO THE ROAD.

When the friction between the tyre and the road is reduced however -
in icy conditions for instance - then you suddenly see the truth of
your statement. The internal application of the brake has no useful
effect.