How do you get rid of enlarged, or large gathering of blood vessels in the eye safely?

Cause? These should be examined first. Could be some sort of hemangioma, sentinel vessel leading to a cancer, increased episcleral venous pressure, infection, inflammation, allergy, etc.
Exam first. The primary goal is to determine why there is an enlarged or engorged group of vessels on the eye. Is there an underlying tumor with a sentinel feeding vessel or is there a pterygium? If no cause is found and the blood vessels are unsightly, excision with cautery usually proves to be successful.