Learn Polish: Marchewka jest pomarańczowa.

It is an adjective describing marchewka, which is a feminine noun. In Polish, adjectives must match the noun's gender, so the base dictionary form pomarańczowy changes its ending to the feminine -a.

Doesn't the verb jest (is) require the Instrumental case? Why isn't it pomarańczową?

The verb być (to be) only forces the Instrumental case when the following word is a noun (for example, Ona jest kobietą - She is a woman). When you describe something using only an adjective, like the color here, it stays in the basic Nominative case.

Is the word for the color orange the same as the fruit?

No, they are different words! The color is an adjective: pomarańczowy (which becomes pomarańczowa here). The fruit is a feminine noun: pomarańcza. To say "An orange is orange", you would say: Pomarańcza jest pomarańczowa.

I sometimes see the word marchew for carrot. What is the difference between marchew and marchewka?

Marchewka is technically a diminutive, but it is the standard way to say "a carrot" in everyday speech and when cooking. Marchew is generally used as a mass noun to talk about the crop in agriculture or as a general category.