"He stopped the car."
Translation:O, arabayı durdurdu.
Araba durdu. The car stopped. Afterwards, the car stood still. The subject of the verb is the one that is standing afterwards.
O, araba durdurdu. He stopped the car. Afterwards, the car stood still but perhaps he is still walking around after having thrown a big rock onto the road that stopped the car. The subject of the verb ("he") is not the one that is standing afterwards.
Or consider other causative verbs:
The door opened. Afterwards, the door is open.
He opened the door. Afterwards, he is not open! (At least, I hope not.) Only the door is.
"durmak" is "stop" when the subject stands still afterwards.
"durdurmak" is "stop" when the object stands still afterwards, i.e. when you "do stopping" to something.
arabası is the third person singular possessive form: "his/her/its car"
arabayı is the accusative case: "the car", as a direct object
For nouns that end in a consonant, the two are the same (e.g. evi "his/her/its house -or- the house (object)") since the ending is -İ in either case, but for nouns that end in a consonant, the endings for the two purposes are distinct.
Arabayı cannot possibly be the subject because it has the accusative ending -yı. As a subject it would have to be araba.
A lot of people seem to be under the misunderstanding that the accusative case simply means "the". It does not. It indicates a direct object. Unlike in most other languages that have an accusative case, the Turkish accusative case is only used for a DEFINITE DIRECT OBJECT, not an indefinite direct object, not a definite or indefinite subject.
Araba durdu = A car stopped. / The car stopped.
Arabayı durdurdu. = He or she stopped the car.