How do I get rid of a busted blood vessel in my eye?

Time. Unfortunately, about the only thing you can do is wait this one out. You may want to use eyedrops, such as artificial tears, if you have an itchy feeling in your eye. Beyond that, it will take 10 to 14 days for the blood to be reabsorbed.
Time. There is nothing you can do except avoid blood thinners and wait.