I agree with all the answers above.
To give a bit more emotional depth to the answer, to me (a native English speaker), reading the quoted drama, it sounds like the the brother is saying "I'll see you later" but knows that he won't. So he's saying it as a way of avoiding saying "goodbye" which sounds more final.
In that way, it's more sad than saying "I'll miss you" since it acknowledges that his parting will hurt them both, without saying it literally. All the sadness is in what they're NOT saying, not what they ARE saying.
It's interesting to note that in most other European languages, the word for goodbye can be literally translated as "until the next time we meet" (au revoir, auf wiedersehn) but in English our word has no such connotation.
(As an aside, the word "goodbye" evolved from the phrase "God be with you".)