Well, generally speaking, when you see the veins in your eyes, it may means that your eyes are irritated by something. And according to some researches, the main cause of the appearance of red veins in the eye is the irritation, dilation, and swelling of the blood vessels within. That is why you can see veins in your eyes. For example, Dry air can cause irritation of the eyes which often results in prominent red veins in the eyes. Also, fatigue and stress can be the causes too. So you need to go and see the eye doctor and he will have a check for you.