Learn Finnish: Tämä auto maksaa paljon.

It actually did change, but the result looks identical! Maksaa is a Type 1 verb. To get the "he/she/it" form (since the car is an "it"), you take the stem maksa- and double the last vowel. That makes it maksaa. It is just a coincidence that it looks exactly like the dictionary base form.

I learned that maksaa means "to pay". How do I know it means "to cost" here?

You can tell from the context and the subject of the sentence. If a person is the subject, maksaa usually means "to pay". Since a car (auto) cannot pay for things, the verb translates to "to cost" in this situation.

Could I just say Auto maksaa paljon to mean the same thing?

Not quite. Auto maksaa paljon means "The car costs a lot" or "A car costs a lot". Adding the demonstrative word tämä (this) specifies that you are talking about this exact car right in front of you.

Does the word paljon need any special endings here?

No, paljon stays exactly as it is. Here it acts as an adverb meaning "a lot" or "much", describing how much the car costs. Adverbs like this do not take extra grammatical endings.