내리뜨다 means "look downward" or "drop one's eyes." The opposite of 내리뜨다 is "치뜨다," which means "look upward." The prefix 내리- means "downward," the prefix 치- means "upward," and the verb 뜨다 means "open (one's eyes)"; therefore, 내리뜨다 literally means, "open one's eyes downward," and 치뜨다 literally means "open one's eyes upward."
I find 내리뜨다 and 치뜨다 a little strange because how does one open one's eyes downward or upward? "Upward" and "downward" imply movement, but the act of "opening one's eyes" happens too quickly to give a sense of movement. Why not just say 내리보다 or 치보다 since 보다 means "to see" or "to look," which is what 내리뜨다 and 치뜨다 seem to be saying? Actually, there are two words that use 보다 that seem similar to 내리뜨다 and 치뜨다. The two words are 내려다보다 and 쳐다보다.
내려다보다 means "look down at," and 쳐다보다 means "look up at," The literal meaning of 내려다보다 is "lower and look," and the literal meaning of 쳐다보다 is "raise and look," which seem to imply that there is more movement involved with 내려다보다 and 쳐다보다 than with 내리뜨다 and 치뜨다.
So what is the difference between 내리뜨다 and 녀려다보다 and between 치뜨다 and 쳐다보다?
I think 내리뜨다 is used when a person has downcast eyes, but is not really looking at anything in particular. The person may be in deep thought or may be trying to avoid eye contact with someone. On the other hand, 내려다보다 is used when a person is looking down at something to see what it is or watch what is happening. Likewise, 치뜨다 is used when a person has upcast eyes, but is not really looking at anything in particular. The person may be looking up while trying to remember something. 쳐다보다, on the other hand, is used when a person is looking up at something to see what it is or watch what is happening.
I have not read any usage notes on the above words, but I think I am guessing pretty close to the differences in meaning.