When you belch you release air and excessive air in your stomach can cause pain. Therefore belching to reduce air the air in your digestive area can relieve pain. I'm not sure why it would be associated with lower right side back pain though since the excessive air, often due to 'air swallowing' or acid reflux, is more likely to occur in your stomach area.
Gas buildup in your intestines can be associated with pain which might be relieved when you belch. There are a number of abdominal conditions that can cause excessive gas including eating large amounts of fatty food or eating foods that your body has difficulty digesting.
You'll have to see a doctor to get this sorted out. Good luck!