Learn Dutch: Gelukkig blijft de koers vandaag stabiel. - Fortunately, the rate remains stable today.

Because the sentence starts with the adverb Gelukkig (Fortunately). In Dutch, putting anything other than the subject in the first position triggers inversion, meaning the verb blijft must take the second position and the subject de koers immediately follows it.

What does koers mean in this context, and are there other common meanings?

Here, koers refers to a financial rate, like an exchange rate or stock price. It can also mean a heading or direction (like a ship changing its koers), and in Belgium, it is commonly used to refer to a professional bicycle race.

Why is vandaag placed before stabiel?

In Dutch sentence structure, expressions of time like vandaag generally appear as early as possible after the verb and subject. The adjective stabiel describes the state of the subject, so it functions as a predicative adjective and naturally belongs at the end of the clause.

This lesson is about the fluctuating structure dan weer ... dan weer. How would I use that to describe the exact opposite of this sentence?

If the rate were fluctuating instead of staying stable, you could use the lesson's target structure to say: De koers is dan weer hoog, dan weer laag (The rate is sometimes high, sometimes low). This perfectly contrasts with the stable situation described here!