Learn Russian: Они неделями беспрерывно спорили.

Here, the bare instrumental plural неделями means "for weeks on end." This is a special use of the instrumental case to show an extended, uninterrupted duration of time or something happening iteratively over a long period. You will also see this pattern with words like часами (for hours) and годами (for years).

Could I just say много недель (many weeks) instead of неделями?

You could, but неделями adds a stronger emotional emphasis. It doesn't just state a large quantity of weeks; it conveys the feeling of "for weeks and weeks," emphasizing how tiresomely long the continuous arguing lasted.

Why is the imperfective verb спорили used here instead of the perfective поспорили?

The imperfective спорили is required because the sentence focuses entirely on the ongoing process and extended duration of the action, as clearly indicated by неделями and беспрерывно. The perfective поспорили would mean they had a single, completed argument.

How is the adverb беспрерывно (continuously) formed?

It comes from the root -рыв- (related to tearing or breaking). You likely already know the noun перерыв (a break or pause). Combined with the prefix бес- (without), беспрерывно literally translates to "without a break."