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: 
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> (1) chyotto matte tte  
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>  (2) why does the tte mean "... and I'll be back shortly". 
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(1) ちょっと待｛ま｝ってって  

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「ちょっと待｛ま｝って」って  

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「ちょっと待って（ください）」って  

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「ちょっと待って（ください）」と  

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「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？）  

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「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。  

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(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"? 
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chyotto matte X  

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ちょっと待って X  

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「ちょっと待って Ｘ」  

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「ちょっと待って、 Ｘ（だから）」  

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「ちょっと待って（ください）、Ｘ（だから）」  

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