Learn Swedish: Äpplet är stort.

Because äpple is an ett-word. In Swedish, when an adjective describes a singular ett-word, you must add a -t to the end of the adjective. Therefore, stor (big) becomes stort.

If I wanted to say 'The car is big', would I also use stort?

No, because bil (car) is an en-word. When describing a singular en-word, the adjective stays in its base form without the extra -t. You would simply say Bilen är stor.

Why is it just äpplet instead of ett äpplet to mean 'the apple'?

In Swedish, you don't use a separate word in front of the noun to say 'the' in this type of sentence. Instead, the article attaches to the end of the noun. Ett äpple means 'an apple', and changing the ending makes it äpplet (the apple). Saying ett äpplet would be like saying 'an the apple'.