I have heard that competence is a key issue in the Regulations. Why is this?

Competence is about being able to do your work safely, not endanger others, and meet the legal health and safety requirements. The Regulations emphasise competence because it is generally recognised that competent people are safer.
The duties in the CDM Regulations work both ways. People making appointments have to take reasonable steps to make sure that those appointed are competent for what they are expected to do. Likewise, those accepting such appointments should only do so if they are competent to undertake the activity. The CDM 2007 ACoP (HSE publication L144) provides practical guidance to help people to assess competence. The advice given in the ACoP will make the assessment of corporate and individual competence easier.