It wasn't that they were against private property because they thought it was sinful, but rather that they thought it was sinful because they were against it*.
To understand this, its probably best to start by talking about common land and enclosures.
Under traditional English Common Law, tenants had certain usage rights on common land, even though technically they did not own the land. These included fishing, grazing, farming, mining. All within strict limits, of course, but they did have a right to use the land to support themselves and their families. This was a pretty useful setup in an era where all land was technically owned (because Feudalism), but England had a lot of untended land laying around.
However, as the population grew, that problem went away, and land in the commons started to become really valuable. As a result, landowners started using the Enclosure process to take their land out of the commons and reserve it for their own exclusive use. While personally lucrative, this turned all the former tenants into the era's equivalent of homeless migrant laborers. As you can imagine, these people were not pleased with their change in status. There were also way more of them than there were landowners, so this led to a lot of social unrest.
This is where the Diggers came in. Simply put, they wanted to undo the enclosures. They were essentially a social reactionary movement. They wanted to find a way to extend the old system of Common Land into the new era. That they searched through the Bible for religious justification should surprise no one. Being a rather large diverse book, the fact that they found passages backing them up should also surprise no one. But chiefly they were just about going back to the old ways of doing things.
* - This kind of situational morality happens far more in history than one might naively think.