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".
> 
> (1) chyotto matte tte  
> 
>  (2) why does the tte mean "... and I'll be back shortly". 
> 
> 
> 

(1) ちょっと待｛ま｝ってって  

↓  

「ちょっと待｛ま｝って」って  

↓  

「ちょっと待って（ください）」って  

↓  

「ちょっと待って（ください）」と  

↓  

「ちょっと待って（ください）」と（私｛わたし｝が言｛い｝ってるのに、あなたはなぜ待ってくれないの？）  

↓  

「ちょっと待って（ください）」と（私が言ってるのに、あなたはなぜ待ってくれないの？）  

「I'll be back shortly」って思｛おも｝ってるのに。  

↓  

(2) Why don't you wait for me a moment in spite of my saying "Please wait a moment"? I'm thinking "**I'll be back shortly.**" 

If "(1) chyotto matte **tte**" is "(1)' chyotto matte **te** ちょっと待って**て**", it is a short form of "ちょっと待って**いてください**", which implies that "すぐ戻りますからちょっと待っていてください *Would you wait a moment, because I'll be back shortly?*" 

The second て in ちょと待って**て** is "an auxiliary indicates continuing action" according to Jisho.org here, so the phrase means "Keep waiting for a moment.​" 

> 
> chyotto 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"? 
> 
> 
> 

chyotto matte X  

↓  

ちょっと待って X  

↓  

「ちょっと待って Ｘ」  

↓  

「ちょっと待って、 Ｘ（だから）」  

↓  

「ちょっと待って（ください）、Ｘ（だから）」  

↓  

"Wait a moment, please, because I'll do X".