Well, the first thing you need to know is that mistake x for y is an idiom. So answer choice (A), which offers us mistook the sound...as the blast... is unidiomatic.

This principle will also let us eliminate (D) and (E).

Answer choice (B) doesn't make any sense. How can someone mistake a car for the blast/sound of a gunshot?

Answer choice (C), the best answer, offers us the idea that Michelle mistook the sound of a backfiring car... for the sound/blast of a gunshot.

I believe that the above question was inspired by this real GMAT question:

In the mid-1960's a newly installed radar warning system mistook the rising of the moon as a massive missile attack by the Soviets.

(A) rising of the moon as a massive missile attack by the Soviets
(B) rising of the moon for a massive Soviet missile attack
(C) moon rising to a massive missile attack by the Soviets
(D) moon as it was rising for a massive Soviet missile attack
(E) rise of the moon as a massive Soviet missile attack