Here are the logical ways to explain the misconception of injury being less common with bodyweight training:

1. Progress with BW is slow as crawl. All things being equal, lifting will improve strength much faster, leading to higher loads sooner. Higher loads always mean more injury potential. Faster progress is not magic - it's possible because all major muscle groups can be loaded precisely and gradually with weights as opposed to BW movements. You can take things as slowly as you want.

2. With BW, some movements simply cannot be loaded properly. For example there is nothing you can possibly do with just your body weight to put any serious load on your legs or low back. This already cuts one's potential for injury in half at the cost of neglecting the development of the lower half of your body. So instead of taking things slow with your barbel squats, you are stuck at 1-leg BW squats forever, which are only hard because of balance/stretching issues and are fairly trivial strength-wise.

3. With free weights, it's easier to be stupid and bite more than you can handle.

As I said, I've done both for enough time to appreciate pros and cons. There are reasons to practice BW training, but lower injury changes is not a good one.