How do you treat oral gum swelling at home?

In addition to the other advice, you can use a teaspoon of salt in warm water as an effective antiseptic mouthwash. 

You might have a piece of food stuck under the gum, so I would clean it out with a toothpick and floss first, even if it bleeds, so you know it is clean, and then use mouthwash and/or salt water. 

Mouthwash and gentle brushing. If the swelling increases then you may have a tooth abscess which requires the correct antibotics asap.

Medicated mouthwash and avoid  your gum when brushing.

Ideally, you see a dentist, but if you absolutely cannot, you step up your oral hygiene big-time. You brush after every meal, using a soft toothbrush, not medium. You brush your gums and tongue. You use mouthwash once a day. You floss in the morning and after the last night-time brushing.

Lost of gum swelling will resolve if you get real serious about brushing and flossing. But if the swelling is a dental abscess, it won't help. (It also won't hurt, and when you do see a dentist, he or she will be able to tell you take good care of your teeth.)

Do you have a water flosser?  An electric tooth brush? Paste that proved for good gum health, floss that actually does something besides go between the teeth? Mouthwash that is decent on gum disease?  Because if you have swollen and or bleeding gums, you got issues.

you don't...............