Learn Dutch: Het weer is niet constant: dan weer regent het, dan weer schijnt de zon. - The weather is not constant: sometimes it rains, sometimes the sun shines.

It is a fixed phrase used to describe rapid fluctuations or alternating states. While dan weer literally translates to 'then again', using it twice like this means 'now [this]... then [that]' or 'sometimes [this]... sometimes [that]'.

Why do the verbs come before the subjects in regent het and schijnt de zon?

This is caused by inversion. Because both clauses start with the adverbial phrase dan weer, the verb must come in the second position, pushing the subjects (het and de zon) to the position immediately after the verb.

Could I just use soms ..., soms ... instead of dan weer ..., dan weer ...?

Yes, you could say soms regent het, soms schijnt de zon. However, dan weer ... dan weer specifically emphasizes a restless or unpredictable switching between two states, which perfectly connects to the first part of the sentence: Het weer is niet constant.

Do I need a conjunction like en (and) between the two dan weer clauses?

No, you do not need a conjunction. The correlative structure dan weer ..., dan weer ... links the two clauses together naturally, so a comma is all the punctuation you need.