The contemporary response to a request, not a problem or no problem, has spread from American English to all of the European languages, e. g., German kein Problem, French pas de problème(s), (Serbo-)Croatian nema problema, Bulgarian няма проблем, Russian нет проблем, etc. Note, however, that the word “problem” is calqued in the plural rather than the source language’s singular.
As they say in Russian, Спрашивается ‘[literally] one asks’, or Why? More to the point, why do people habitually resort to this mindless formula? Apropos, here’s a bit of text from an exchange between a customer and a waitress (native speaker of American English, ca. 25 years old):
Waitress: “Would you like to see a dessert list?”
Customer: “Actually, I’d just like a double decaf espresso.”
Waitress: “Not a problem.”
Now, such a quotidian exchange might need no interpretation, being utterly straightforward. But the question arises nonetheless, why the waitress didn’t just say “Certainly, sir,” or “Of course,” or “Right away.” Why just “Not a problem?”
A literalist approach might make one surmise that the waitress actually wanted to convey the idea that, whereas some restaurants had no espresso machine, hers did, hence making a demitasse of double decaf espresso would not constitute a problem. But this would be tantamount to clobbering linguistic gnats with a psychological trebuchet. Her réplique was, after all, just a mindlessly spontaneous resort to a token of language use tout court, a cliché and nothing more. But this description is too facile, for the following reason.
Every utterance involves a choice by the speaking self from among a repertoire of contextually equipossible variant words or expressions signifying the same general meaning, and the range of possibility includes inventories of clichés as well. The waitress made a choice, which means that she privileged an expression token involving the word problem rather than the alternatives. The upshot is the involvement of an IMPLICIT HIERARCHY, where rank relations among possible variants always imply the presence of VALUE STYLE as a necessary frame for all uses of language, no matter how humble or mundane.
[Addendum: Patrick Honan points out (viva voce) that “no problem” is commonly used nowadays instead of “you’re welcome” as a retort to “Thanks.” One could, of course, construe this usage as a shorthand for “Whatever you’re thanking me for was not a problem (for me).”]