The Two main uses of the Ujjayi Breath in Asana Practice
1) The use of the body's fire in Hatha Yoga
According to scholars of yoga scriptures, traditional asana schools fall into two classes : the Hatha yoga system and the Patanjala yoga system. These are akin to the Yang and the Yin systems, respectively. Most asana schools today are a mix of these two systems. The Ashtanga Vinyasa style of yoga, devised by Shri T. Krishnamacharya and popularized by Shri K. Pattabhi Jois, is an example of a Hatha Yoga styled practice due to the strong emphasis on the Ujjayi breath and the use of the bandhas, to generate and channel the heat, respectively. On top, since it was taught to agile teenagers by Krishnamacharya, it had a component of flow through the use of vinyasas. It has to be noted that even those modern day schools which go by the name Hatha yoga, do not strictly follow the Hatha yoga system in terms of kindling and maintaining the body's fire.
A Hatha Yoga practice uses the body's fire to cleanse the body as described in the verse below from the Gheranda Samhita.
The fire also makes the body malleable even if one is not innately flexible. This fire is considered divine because it is what sustains the life force, Prana. Another name for the Jatara Agni is Vaishvanara. In chapter 15 of the Bhagavad Gita, Bhagavan Krishna says,
Becoming the digestive fire, Vaishvānara, I reside in the bodies of all beings. Associating with Prana and Apana, I digest the four types of food (1. Bhojya - Foods that are chewed, such as bread, chapatti, etc. 2. Peya - These are mostly liquid or semi-solid foods which we have to swallow or drink, such as milk, juice, etc. 3. Kośhya - Foods that are sucked, such as sugarcane. 4. Lehya - This includes foods that are licked, such as honey, etc.)
2) Using the Ujjayi breath to kindle the Agni
The Agni, described above, is kindled using the Ujjayi breath and is fanned throughout the body using the bandhas. The following article describes how the breath, bandhas and the Agni (fire) enhance one's yoga asana practice.
A material analogy to a visceral feeling of the use of bandhas to fan the fire generated by the breath is shown in the video in this slide that shows the blacksmiths bellow that stokes and fans the fire.
3) Using the Ujjayi breath for gaining traction in difficult poses
The second use of the Ujjayi breath is to gain traction while trying to do a complex pose. When the body is contorted, it is naturally difficult to take a long inhale and exhale. This is where the Ujjayi breath comes in handy. Since it has a component of friction and therefore control, it is much easier to do a long exhale and inhale using a Ujjayi breath as compared to a long subtle breath (Patanjali's Dheerga Sukshmah in Sutra 2.50 from the Yoga Sutras) in a difficult pose.
This is best illustrated using Kapotasana, a deep backbend, which is notoriously difficult for those who do not have a flexible lower/middle back, which is most people who sit for long periods of time at a desk. Things are made more difficult by the backbending nature which is inherently disorienting. By using a measured breathing with plenty of traction provided by the Ujjayi breath, the mind is able to focus instead of going haywire. At time 5:00 of the video below one can get a feel for how the Ujjayi breath can be used in conjunction with the crawling movement of the hands on the mat (a term jokingly called "The Kapotasana Pilgrimage") to maintain focus