De auto is kleurrijk.

Because auto is a de-word in Dutch. In the singular, Dutch nouns take either de or het, and auto belongs to the de group.

• de auto = the car
• de auto's = the cars

A useful extra detail:

• the basic noun is de auto
• the diminutive is het autootje = the little car

So the article changes with the form of the noun, not with the meaning car itself.

Because kleurrijk comes after the verb is.

In Dutch, adjectives used after a linking verb such as zijn usually stay in their basic form:

• De auto is kleurrijk.
• De auto is mooi.
• De auto is groot.

But when the adjective comes before the noun, it usually takes -e:

• de kleurrijke auto
• een kleurrijke auto

So:

• De auto is kleurrijk. = adjective after is → no -e
• de kleurrijke auto = adjective before auto → -e

Yes, in this simple sentence it is very similar to English:

• De auto = subject
• is = verb
• kleurrijk = adjective / complement

So the structure is basically:

• subject + verb + adjective

Just remember that Dutch has a broader rule that the finite verb usually comes in the second position in a main clause. This sentence follows that rule naturally.

Kleurrijk is an adjective. It describes a quality of de auto.

It is built from:

• kleur = color
• rijk = rich

So literally it means something like color-rich, which is why it means colorful.

It can be used:

• literally: a colorful car, painting, dress
• figuratively: a colorful personality, story, or past

In De auto is kleurrijk, it is being used literally.

Very roughly:

• auto → OW-toh
• kleurrijk → roughly kluhr-rike, with the second part sounding roughly like English bike after an r

A few helpful notes:

• au in auto sounds roughly like the vowel in now
• eu in kleur has no exact English equivalent
• ij in rijk sounds roughly like the vowel in my

These English spellings are only approximations, but they are a good starting point.

You move the verb to the front:

• Is de auto kleurrijk? = Is the car colorful?

This is the normal pattern for a Dutch yes/no question:

• statement: De auto is kleurrijk.
• question: Is de auto kleurrijk?

You use niet:

• De auto is niet kleurrijk. = The car is not colorful.

Here niet negates the adjective kleurrijk.

The plural is:

• De auto's zijn kleurrijk.

What changes:

• auto → auto's
• is → zijn

What stays the same:

• kleurrijk stays kleurrijk after zijn

So:

• singular: De auto is kleurrijk.
• plural: De auto's zijn kleurrijk.

The apostrophe in auto's is part of the normal plural spelling for many Dutch words ending in a vowel.

For most learners, the important thing is simply that auto is a de-word.

In modern Dutch, learners usually focus on:

• de auto
• de auto's

rather than worrying too much about older masculine/feminine distinctions. So at beginner level, the best habit is just to memorize auto with de.