Learn Russian: Он много работает для того чтобы купить машину.

Because the person working and the person buying the car are the exact same ('He'). You only use чтобы plus the past tense when there are two different subjects, like if he were working so that someone else could buy a car. Since he is doing both actions himself, Russian just uses the infinitive.

What is the difference between для того чтобы and just using чтобы?

They both mean 'in order to.' The longer phrase для того чтобы is a bit more formal and puts heavy emphasis on the goal. In everyday conversation, Russians often shorten this and just use чтобы by itself.

Why use the perfective verb купить here instead of the imperfective покупать?

He has a specific goal to complete a single, successful action: buying one car. The perfective aspect (купить) focuses on achieving that final result. If his goal were an ongoing or repeated action, like buying cars regularly for a business, you would use the imperfective покупать.

Just a quick refresher: why does машина change to машину at the end?

It is the direct object of the verb купить (it is the thing being bought), so it needs to be in the accusative case. Feminine nouns that end in -а change their ending to -у in the accusative.