Learn Turkish: Bu efsane nasıl başlamış?

The suffix -mış is the inferential or reported past tense. It is used here because the speaker wasn't there when the legend started. Legends, myths, and historical events that we didn't personally witness almost always take -mış rather than the direct past tense -dı.

How do we know to use -mış instead of -miş or -muş for this verb?

This is due to 4-way vowel harmony. The root of the verb is başla- (to start), and its final vowel is a. Under the 4-way rule, roots whose last vowel is a or ı will take an ı in the suffix, making it -mış.

Why is the question word nasıl placed right before the verb?

In Turkish, the most important information in a sentence—the part you are focusing on—is usually placed right before the verb. Since this is a question asking 'how', placing nasıl immediately before başlamış is the most natural word order.

Can efsane be used for everyday things, or just historical myths?

While it literally means a historical 'legend' or 'myth' (which fits perfectly with the -mış tense), Turkish speakers also use efsane in modern slang exactly like English speakers do. You can use it to describe an amazing party, a great sports player, or a fantastic meal!