A herniated disc is usually an outcome of a chronic degenerative process. That said, positioning can be an aggravating or precipitating cause and deconditioning of core muscles often contributes.
In terms of the mechanics of the process, discs bulge or herniate by being squeezed out posteriorly from between the vertebral bodies, so flexion (forward bending) of the spine is what does it. This is something one should try to minimize on the bike and elsewhere by rotating the pelvis forward and flattening the lumbar spine instead of bending it. Think about the proper form for a starting a deadlift. Doing a deadlift wrong is a perfect way to squeeze out a disc and doing them right is a great way of improving position on the bike and protecting the spine.