"אין בעד מה!"
Translation:You are welcome!
It would literally mean "Nothing" "for" "what"... So in response to "Thank you" it would mean something like "For what?... It's nothing"
Like others have said... It has similar meant to other common English phrases like "don't mention it" or "forget about it" or "don't worry about it" or the closest literally but not so common in English "it was nothing"
If it helps, Hebrew uses a root system which is much more extensive than English. I personally am not good enough to know what the root is here, but I can see that within the word בבקשה at least two other words: בקשה (request) and קשה (difficult or hard). With this in mind perhaps this is closer to saying "to your request?" In other words "(It's nothing,) I'm fulfilling your request." Or more simply בבקשה could mean only please, and is used as "Please, don't thank me."
Literally בבקשה means "in a request", slightly less literally "as a request". I'm quite that the intention, before it was frozen as an idiom, is "you don't have to thank me, I'm asking you to have it (take advantage of it, etc.). Hebrew probably adopted it from German "bitte" (literally "I am asking") or French "Je vous en prie" (literally "I am asking you that").
While you can say "kein Ding" in German, it is very colloquial and not as polite as other options. You would say it with close friends, but in a work place or with strangers you are better off saying "keine Ursache" (most standard) or "kein Problem" (which would be like ain beaya according to an Israeli friend). Just wanted to add, in case someone who is learning German comes across. :)
Aw, shucks is somewhat regional and dated, but is an idiomatic phrase from the American South, "aw, shucks t'wern't nothing" is a politely embarrassed way of receiving a thank you or receiving credit or praise for something. Aw shucks is akin to "Gee, golly". It's just an expression or interjection of bewilderment. Usually it's used nowadays in humorous or sometimes ironic fashion.
Shucks can also be used for mild chagrin or disappointment, or a mild oath. Read some Mark Twain you'll get a sense of it.
bevakasha בבקשה al lo davar על לא דבר ein baad ma אין בעד מה
Can all be used as a reply to תודה. Other Hebrewers here said there is some difference in level of formality. They may be right but IMHO the differences in formality are microscopic... You can choose any and you'll be fine (:
I just posted the link to this whole expression. This sounds to me as:
Ain bead ma.