This can occur from coughing, straining, sneezing, or rubbing the eye. Often, the person is on aspirin or other blood thinners, which makes the tiny ruptured vessel bleed more rather than clot quickly. Generally it is harmless and called a subconjunctival hemorrhage. If your eye hurts, vision is worse, you remember getting hit in the eye, you see flashes or floaters or eye is protruding or bulging out, then you should see your ophthalmologist. Otherwise it will go away in a week or two all by itself. There is no way to make it go away faster. The body will reabsorb the blood just like a bruise.
MJK MD