This is an excellent question. Let me name a few grammatical points that will help you.
First, when you use an article before a comparative, such as in "the more," you are usually setting up a comparison between two comparatives. In short, the change one is linked to the change in the other. In this case, "the more you use computer" is related to "the more you damage your eyes."
Second, the subject is the word, "you" and the verb is "damage," and "use." When you look at it in this way, the sentence can be seen in this way: "You damage your eyes when you use the computer." Also, an article is reqiuired in English before "computer." Use either "the" or "a": "use the computer" or "use a computer."
Third, since you are comparing two actions, one comparisonis not made subordinate from a grammatical point of view. In short, when there is an article before a comparative, then it sets up a comparison between two comparative words.
This is a classic example of a balanced sentence in which the structure of each part mirrors the other--in this case, each part is exactly like the other--and such sentences are often effectively used to emphasize a cause-and-effect relationship (as in your example's case) or to compare or contrast the two elements. Balanced sentences are especially effective because they appeal to the human tendency to like symmetry and balance, both in physical things and statements, and often appear in works or speeches designed to convince readers or listeners to take a particular action.