Question: When someone says: `chotto matte te`, why does the `te` mean "... and I'll be back shortly". What's **the logic** behind it? 

Why aren't there special expressions such as `chotto matte X`, where X could mean "I'll give food to my cat" or "I'll need sign an important document for Toyota" or "I'll fix my laptop which was broken by little sister"? Why come that `te` means **specifically** "...and I'll be back shortly"? And why is it "te"?

Answer: 
* ちょっと待っていてください　-> ちょっと待っていて -> ちょっと待ってて(chotto matte te)
* ちょっと待ってと言っているでしょう -> ちょっと待ってと -> ちょっと待ってって(chotto matte tte)

These two phrase are similar, but not the same.
It's short for ちょっと待っていて, literally "wait and stay there". There's nothing about coming back, but of course it is inferred.