Learn Turkish: Meydan mahşer yeri gibiydi, panikle kaçışan kaçışana. - The square was like a madhouse; people are fleeing in panic left and right.

This uses the 'chaotic abundance' structure taught in this lesson. You take a verb with the subject participle -An and repeat it with the dative suffix -(y)A. Literally meaning 'the fleeing one to the fleeing one', it idiomatically paints a vivid picture of everyone doing the same frantic action all over the place in total chaos.

Why use kaçışan instead of just kaçan? Doesn't kaçmak already mean 'to flee'?

Yes, kaçmak is 'to flee'. The -ış- here is the reciprocal suffix. While it usually means doing an action 'to each other' (like görüşmek, to meet), when added to verbs of motion it means a group of people doing the action in a scattered, disorganized way. Kaçışmak specifically means 'to scatter' or 'to flee in all directions'.

What does mahşer yeri literally mean?

Mahşer refers to the grand gathering of souls on Judgment Day in Islamic theology. So a mahşer yeri (place of gathering) is a common idiom used to describe an intensely crowded, chaotic, and loud place, much like saying 'a madhouse' or 'a zoo' in English.

The first half is in the past tense (gibiydi), but the translation says 'people are fleeing'. Does kaçışan kaçışana force the present tense?

No, the -An ... -AnA structure lacks a tense of its own. It acts as a descriptive phrase that inherits its time frame from the main context. Since the square 'was' (gibiydi) a madhouse, the fleeing also happened in the past. The English translation just uses 'are fleeing' to capture the vivid, immediate feeling of the Turkish phrase, but 'people were fleeing' would also be correct.