My blood test for thyroid came back negative but I have visible throat swelling. What could it be?

Get an ultrasound. The blood test does not rule out the presence of thyroid nodules/goiter/inflammation. It just tell you that currently, the throat swelling is not disrupting the ability of the thyroid to produce hormone. The best way to look at the physical aspect of the neck is through thyroid/neck ultrasound. I agree with the need to measure thyroid antibodies to see if you have autoimmune thyroid disease.
Possibilities. Most folks with hashimoto's, especially early, have normal thyroid function tests though antibodies against thyroid proteins are common. People with nodular goiter, which is even more common, usually have both tests normal. Agree that ultrasound is a reasonable next step.
Goiter. Enlarged thyroid gland is the likeliest cause. Thyroid function is different than inflammation. You likely have hashimotos disease which can be diagnosed by measuring thyroid antibodies in the blood. Get an ultrasound of your neck.
It can still. Be a goiter with normal thyroid tests, and if not, a lymph node or branchial cleft cyst would be possiblities.