In many ways bodyweight training is safer than lifting weights. You don’t have compressive forces pressing down on your spine. You don’t have the problems of getting weights into position. You don’t have the problem of being pinned under heavy loads. You don’t have the problem of losing control of a heavy object mid-exercise.*
However, trust me, you can still hurt yourself using bodyweight exercises.
No training method is risk free!
When you start using more advanced bodyweight progressions the difficulty stems from putting your joints and muscles into disadvantageous positions. This is a good recipe for muscle or connective tissue strain if you don’t approach your training with a bit of common-sense. It is also why I no longer recommend gymnastic exercises or advanced calisthenics for people unless they have a burning desire to master a particular bodyweight skill.
Even with joint-friendly exercise choices the principles of sensible workout volumes, controlled rep speed, good technique and sufficient recovery remain paramount for staying strong, healthy and injury-free whilst using bodyweight exercises.
*Admittedly many of these problems are eradicated by well-designed machines