How can breathing exercises improve sleep?

Slow, deep breaths have a calming effect on your body and mind, which help you fall asleep. Taking deep breaths helps activate the parasympathetic nervous system, which is responsible for the body's rest and digest mechanism. Mindful deep breaths decrease the heart rate and lower the blood pressure, relaxing the whole body. Being in a relaxed state reduces the secretion of stress hormones, which block melatonin, the hormone associated with sleep. Regular breathwork and mindfulness practices also help reduce your stress level throughout the day, eventually facilitating better sleep at the end of the day.

Deep diaphragmatic breathing is one of the simplest practices you can try for deep relaxation. In deep belly breathing, you focus on expanding the belly as you breathe in and comfortably let the air out. Another technique that famously helps with deep relaxation and sleep is the 4-7-8 technique, where you breathe in for a count of 4, hold for 7 and breathe out for a count of 8. Box breathing or square breathing is another method that calms the mind and reduces stress. Box breathing follows a pattern of inhaling, holding, exhaling and holding, where each step is of equal duration. So, you breathe in for a count of 4, hold for 4, exhale for a count of 4 and hold for 4.

You can find more on deep belly breathing, 4-7-8 technique and box breathing here. 
Another method you can try to fall asleep more easily is the body scan method. It is a mindfulness meditation technique combining breathing and focusing on your body. In this method, you scan your body from head to toe (or vice versa) and concentrate on relaxing each little part of your body one by one. If you feel any tension, you breathe toward that area, making sure you are loosening up all your muscles. You should keep your focus on your breathing and gently direct your attention back to your breath whenever you get distracted.