My tonsils are inflamed and I can't swallow. Are there home remedies that can help?

Salt water. Sounds gross, but gargling with salt water gives a lot of people relief. If you're having that much difficulty swallowing though, you should see a doctor because you may need an antibiotic and other treatment.
Old timey. You can try gargling with salt water or peroxide but, if you can't swallow, things may be more serious than that. Get to your primary doctor asap and see if you have a complicated infection.