Why do we breathe through the mouth in some breathing exercises?

Generally speaking, nasal breathing is healthier than mouth breathing since the nose is better equipped at priming the air for respiration and regulating the oxygen/carbon dioxide balance.

Chronic mouth breathing can lead to issues such as dry mouth sore throat, and nasal congestion. 
However, in some exercises, we breathe in and out through the mouth to intentionally disrupt the oxygen/carbon dioxide balance. This leads to heightened breath awareness. It also helps lengthen the exhales to create a soothing effect or to boost energy, depending on the technique used.

Keep in mind that breathing patterns can be manipulated for short periods to achieve a certain effect. However, breathing habits that deviate from the norm, such as chronic mouth breathing, can lead to many problems.