To understand this, we have to take it step by step. Firstly, if the speed is doubling, then that means kinetic energy increases by a factor of 4, since:
K.E = 12mv2 therefore K.E v2
So 2v leads to (2)2K.E = 4K.E
Now that kinetic energy has increased by a factor of 4, the car has to do more work to slow down to a stop.
Since W = F x d, therefore W d
So 4W leads to 4d
Therefore, braking distance d increased by a factor of 4 as well!