No. Legible refers to how writing appears, as in how easily it can be read or understood. "His handwriting is very legible and understandable." "Your writing is so bad, I don't understand half the words you wrote. It's not legible at all!" Instead, you can say, "This is a clear reason to manage the heritage site, so it can increase people's cultural awareness."

Always put the sentence you are talking about within quotes. Thus your question should have been written like this: Is the word "legible" used correctly in the following sentence: "This is the 'legible' reason to manage a heritage site so it can increase cultural awareness." ? The answer to your question is "YES". "Legible" means "readable". It comes from the Latin verb "legere" = "to read". However, like "read", its use is broader than that, extending to situations that involve no written materials. For example, you could say "His motives are illegible." to express the fact that you cannot figure out what his motives are. In the same way, you could say "I cannot read his motives."