Breathing is important in kendo. Come to think of it, it is generally important. Specifically to kendo however, the way we breathe has an enormous effect on our progress and on the effectiveness of our technique.
The breathing style used in kendo is known in Japanese as Aun no kokyu. We breathe in through the nose hold the air in our abdomen and then expel the air through the mouth. This is the type of breathing associated with yoga and meditation. In kendo however we use aun no kokyu to aid the explosive power of our waza and in concert with our kamae, as a force to repel attacks. Try breathing in as your opponent is moving into striking distance and the chances are that he will attack and you will not be able to resist. If he tries the same approach when you are either holding your breath or slowly expelling air, you become far less vulnerable.
As a general rule we breathe in when we are in safe distance, hold the air in our tanden, expelling some by way of a kakegoe shout in issoku-ito-no maai. We then reserve what is left of our breath until we strike, letting out the remainder as me make our kiai and take zanshin. We only breathe in again after we are back in safe distance. Kendo-no-kata, whilst teaching us many other elements of kendo, gives us a perfect illustration of the correct way to breathe. If we use the first form of the Tachi-no-kata as an example ; both uchidachi and shidachi breathe in on taking jodan, both then hold that breath until they make their respective men attacks, releasing air on their kiai and breathing in after completing zanshin.
In shinai kendo we are in a strong position when we are replete with air or slowly exhaling. So for example with debana men; we breathe in before we make our initial seme, release some breath as we step in, hold our breath in tame and then explode into the technique whilst letting out our kiai. What happens if you run out of air? My suggestion is that you move back to safe distance, breathe in and try again.
There are lots of opportunities to practice correct kendo breathing. One is of course during mokuso, particularly after keiko when we may be out of breath and need to slow things down. The idea is to breathe in quickly through the nose, hold the breath in your tanden for as long as is comfortably possible, then breathe out slowly. Another classic approach is through the practice of kirikaeshi, aiming to complete the first shomen, nine yokomen and the second shomen strike in one breath.
Whichever way you do it, the most important point with breathing is not to stop.