It's used pretty much the same way that الله يسمع منك (allah yesma' minek) "may God hear you" is used. You can reply with this phrase whenever someone says something that you want to happen. You're basically saying that you hope God/heaven will hear what the person just said and make it come true. The phrase is at least used in Syrian, Lebanese, Jordanian, and Palestinian, but I don't know if it's used elsewhere. I have 2 example clips to help with pronunciation and to give some context.
- In this first one the man and the women are complaining about making too much money. The man says that he heard the government is going to lower wages and then the woman responds من تمك لأبواب السماء.
يا ستي طمني بالك. سمعتإلك من مصادر موثوقة أنو في قرار قريب لتخفيض الرواتب (ya sitti tumini baalik. ism'atilik min masaadir mawsoo'a anu fi qaraar 'areeb litakhfeed arawaatib) - Madam, don't worry. I heard from trustworthy sources that there will soon be a decision to lower wages.
من تمك لأبواب السماء (min temek labwaab assama) - From your mouth to the gates of heaven.
Ok, so the guy calls her يا ستي . This is his wife he's talking to. The word ست is more often used to refer to your grandmother or more formally to a woman. Think "my lady".
طمني بالك means "don't worry". Literally, "calm your mind".
مصادر موثوقة is "trusted sources". You'll notice that he pronounces the ث as an س which is very common.
- This second example happens after these guys' team wins in a soccer semi-final. The guy on the left has just said that he hopes the team will have the same results at the next match. The guy on the right replies من تمك لأبواب السماء.