It looks like you have a busted freeWHEEL; not a cassette.

I've seen such failures. Primarily, at the factory, most likely, they failed to insert one ring of tiny bearings. Usually the outboard set. Why do they forget? Don't know. But I've serviced at these two freewheels in the last 2 years with exactly the same problem. These were either a SunRace or a Falcon freewheel. Once was on the road. And I knew they didn't put bearings in the outboard races, because there was no evidence of metal-wear that would be present if the bearings were there.

The lack of the outboard bearings means the body and the cogs have friction. This means the inside body will rub on the freewheel retaining/lock-ring (the one on the face of the freewheel with two holes for a pin spanner). Normally, if both sets of bearings are in, the edge of that lockring (which is reverse threaded) never touches the side of the inner part of the freewheel body. But without that outer row of bearings, the edge touches and over time, when you coast on the bike and it's in freewheel mode, instead of naturally tightening the lockring, the friction on the edge actually loosens the lock ring. So it comes off, and the whole body slides outboard and then you screw your chain alignment and then you break cogs.

Solution?

Get a $10 Shimano 6 speed replacement freewheel from Walmart.com or some other place. Maybe they have free site-to-store shipping. Remove the body stuck on the hub still (big pipe wrench works if you don't mind mangling the fw body).

Or if you have a bag of 120 or so 3/32 inch diameter tiny ball bearings, and the pawls are still functioning, you can grease the FW and body, carefully fill up two rings of tiny balls (about 40-something on the inner ring and 50 something on the other ring), then insert body carefully with pawls pushed in, and then from the other side, thread on the lock ring and tighten nominally with pin spanners.

There is a reason to avoid BSOs. This is one of them.