Sugar is literally evaporated sugar cane juice. You need a lot of cane to make a little sugar because water is heavy. "refinining" is just separating out the molasses to make it white and so the molasses can be sold separately. You only need "heavy industrial processes" to make it profitably at scale.

The refined sugar you buy in the store ('table sugar') is clarified with phosphoric acid and bleached using a number of other chemicals. In addition to this, it goes though a number of other industrial processing steps that you would not be able to perform at home. Hence, it is 'highly processed'.