Purpose is to practice diaphragmatic breathing. Lying on your stomach gives you feedback as to whether your chest or stomach is moving as you breathe. You can do the same thing by sitting, or lying on your back, and putting one hand on your chest and the other on your belly. The belly hand should move and the chest hand should not.
For crocodile breathing, lie on your stomach, relax, and breathe deeply in a way that you feel the sides of your stomach expand. Feel like you're drawing the air down deep into your abdominal cavity, all the way to your pelvic floor. Breathe in slowly, and relax to exhale. Pause for a while between breaths.
The more we practice diaphragmatic breathing, the more it becomes our natural way to breathe all the time. This is the goal. It's healthier, and it helps turn down the stress response. Most of us tend to breathe with our chests and this can have a cascade of negative effects including tension in the neck and traps and remaining in a "stressed" state where stress hormones are on and the sympathetic nervous system is dominant. So, we work to reverse this tendency with diaphragmatic breathing practice.