Pure speculation, but: I imagine the processed nature of the fiberless fruit juice (with all the other ingredients/preservatives/crap they add into it) adds to the cancer risk. I'd also guess that were one to juice the fruit themselves, that would be better than the processed kind, but you're still drinking liquid sugar and bypassing the benefits of ingesting fruit in the first place.

Solid fruits are not linked to cancer, although the pesticides in them can be carcinogenic.

In addition to the fiber, I wonder if it's also a quantity issue. It takes about 4 oranges to make 1 cup of orange juice.

There's often no fiber in "fruit juice". It's just sugar-water derived from fruits instead of sugar cane.