Quote Originally Posted by mercurius
I wonder where this meaning came
from. Is it specific to advertising or it is used in this sense anywhere else?
No, it's not for advertising at all. I don't know where it came from, but it is (or rather was) mostly used in slogans, Soviet (that is political) posters, and so on. For example, there was a famous slogan "Даёшь стране угля!" which meant "Let us miners mine more coal for our country!" ("let's be good workers and be of benefit to our State").

It can also be used when talking about political demands, for example, something like "Даёшь отставку кабинета министров!" (but however, it sounds like being screamed out from a crowd, or like a stylization of that).