Learn Arabic (Modern Standard): هذه هي السيارة.

In Arabic, there is no word for "is" in the present tense. Usually, you just put two words together. But because "the car" (السيارة) has the word "the" (ال) attached, saying هذه السيارة would just mean the phrase "This car". To make it a full sentence meaning "This is the car", we insert the pronoun هي to separate the words and act like "is".

Why do we use the feminine words هذه (this) and هي (she) instead of the masculine ones?

Because the word for car, السيارة, is feminine! You can tell it is a feminine word because it ends with the letter taa marbuta (ة). In Arabic, the demonstrative ("this") and the pronoun ("it/she") must match the gender of the noun they refer to.

When listening to this sentence, why doesn't السيارة sound like "al-sayyarah"?

The letter seen (س) at the start of the word is a "Sun Letter". When you add "the" (ال) to a word starting with a Sun Letter, the "L" sound is skipped. Instead, you double the Sun Letter. That is why it is pronounced "as-sayyarah".

If I just wanted to say "This car", how would I write that?

You would drop the middle word and just write هذه السيارة. By removing هي, it becomes a simple phrase ("this car") rather than a complete sentence ("This is the car.").