Taking big mouth breaths to get oxygen into the lungs for some people may be their way of breathing, you may also do this when you are stressed, or doing some sort of high intensity activity. This can change the CO2 levels in the body as the relationship between the Oxygen and CO2 which will have been changed as you are now breathing in more oxygen than required. Therefore you could now have lower levels of CO2 in the tissues so the oxygen you just breathed in through the mouth won’t be as successful in getting around to the tissues, as we need the CO2 in the tissues to trigger the breathing mechanism, which can then lead to you feeling fatigued.
By breathing through your nose, you can help with diaphragmatic breathing, getting nitric oxide into the lungs and regulating the oxygen flow into your lungs and around to the tissues. This may help to reduce fatigue in relation to better breathing. However, when you are exercising hard, you may need to breath through your mouth as well, but if you are breathing through your mouth when you are exercising at a lower level of effort, it may change the oxygen/CO2 relationship and supply less energy to the muscles needed for exercise.
This is a really just a basic overview of a complicated subject, hopefully I have explained it in a manner that you can understand?