I hope this helps - consider yourself as you perform a push up to help visualize the situation and biomechanics.
Locking the elbows at 90 degrees during pushups can lead to joint injury because "peak axial forces on the elbow joint average 45 percent of bodyweight" (An et al., 1992). In other words, your joints are taking on a large amount of your body weight rather than your muscles. This wears downs the joints and can lead to ...
This detailed solution discusses why locking the elbows at 90 degrees during push ups can lead to a joint injury. APA references are included.