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: 
ちょっと待ってて (chotto matte te) literally means "Keep waiting for a while (please)." That て (te) at the end does **not** mean "I'll be back shortly", at least grammatically.

ちょっと (chotto) just means "for a while", "a little", etc. 待ってて (matte te) is constructed as follows:

* **待つ** (matsu): simple intransitive verb, "to wait"
* **待って** (matte): te-form of 待つ
* **待っている** (matte iru): 待って followed by a [subsidiary verb いる](https://en.wikibooks.org/wiki/Japanese/Grammar/%EF%BD%9E%E3%81%A6%E3%81%84%E3%82%8B) to denote ongoing action. "to keep waiting", "to be waiting"
* **待ってる** (matte ru): [い in subsidiary-いる can be omitted](https://japanese.stackexchange.com/a/18159/5010) in casual speech. "to keep waiting".
* **待ってて** (matte te): te-form of 待ってる. [Sentence-end te-form can express a request](https://japanese.stackexchange.com/q/12026/5010). "Keep waiting!"

Similar examples include 見てて ("Keep watching (please)!"), 黙ってて ("Keep quiet (please)!"), etc.

As you can see, grammatically speaking, there is no "I'll be back" in the original sentence. Perhaps you saw someone's free translation. Note the important difference between 待って and 待ってて; the former is "Wait!", whereas the latter *sounds like* "Wait a moment" or "I'll be back soon".
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.