During ventricular filling, the papillary muscles are relaxed and the AV valves are opened by the pressure gradient between the atrium and the respective ventricle.
This illustration may be helpful: http://www.youtube.com/watch?v=rgY7Ic_9K0M
At the start of ventricular systole, contraction of the papillary muscles tenses the chordae tendineae, and causes the AV valves to shut (S1). This minimizes regurgitation of blood into the atrium during ventricular contraction.
When both papillary muscles are damaged in the left ventricle (such as by papillary muscle infarction), the clinical consequence is typically mitral regurgitation.
Poorly contracting papillary muscles (in the left ventricle) may allow the cusps of the mitral valve to prolapse into the left atrium during left ventricular systole. When the leaflets of the valve prolapse into the atrium, the inner margins of the cusps are further separated making regurgitation more likely.
Of course, additional factors may contribute to mitral regurgitation depending on the presence of other predisposing factors. For example, an abnormally dilated left ventricle may widen the mitral valve annulus, and regurgitation can occur in the absence of pathology in the mitral (or tricuspid) valve per se. This phenomenon is called “functional” regurgitation.