I’m better off without you.
I’ve been trying to get a handle on “dran” for a while. I get that “Ich bin dran” means “It’s my turn,” but “dran” clearly doesn’t mean "turn,’ right? So, beyong "turn,’ I sort of get how it operates, but not how to define it. Suggestions?
Well, literally “dran” kind of means “attached” or “at” to the point of touching. Also, “dran” is a contraction of “daran” which is basically “da + an”, so there’s also a “there” hidden in there. Similar to english “thereof”, “herewith” and words with that pattern, so you could say it’s literally “thereat”.
Beyond that, the two main use cases are for “it’s your turn” and “you’re better off”, as you see.
For the “dran sein” meaning, it’s actually short for “an der Reihe sein” - so you’re “at the turn”.
For the “off” meaning, it’s just idiomatic, and I can’t really tell you what you’re supposedly “attached” to. But I guess it’s the same in english with the “off”, like off of what? (*)
“dran” is most often used with “besser” or “schlechter” (better/worse off), but also with “arm”: “ich bin arm dran”, which is like “so poor that you need to be pitied”.
And there’s a joking phrase:
Lieber arm dran als Arm ab. - Better to be poor than to lose an arm.
This is of course playing with the fact that “arm” and “Arm” are homophones and “dran” and “ab” are opposites.
Edit:
(*) I guess the english “off” refers to how you end up after you got out of (off) a difficult situation?
Then the german “dran” might refer to that same situation, or life itself, and the mindset is that you’re still in that situation. 
@Spielratte check out this dictionary entry: dict.cc dictionary :: dran sein :: German-English translation
To be “gut dran” means “to be well off”. To be “besser dran” means to be “better off”.
I can’t explain why the expression is the way it is, or where it comes from. Similarly as you presumably can’t explain the origin of “to be better off”. Why “off”? “Off” what? It just is that way. I wouldn’t think excessively about it. Does it really matter? I think not.
There is something called “diminishing returns” and you don’t gain much by thinking it through. This “dran sein” is an instance where I would just accept it as a fixed expression (to be “besser dran”, to be “better off”) and learn it by heart.
“Ich bin dran” can mean:
Paired with a word like “better” (Ich bin besser dran (als du/ohne dich)) can mean:
Here’s the dictionary entry for “dran” by itself (rather than “to be ‘dran’”), for other uses of the word “dran” (some of them already mentioned by @pitti42): dict.cc | dran | English Dictionary
“thereon”, I guess, but that means something else in English now.
Why would “You’re up!” would mean “Du bist dran.”? Just does.