mame (ni) expresses the way in which one seriously works hard at something without complaining about the task. Often you may translate it as "diligent(ly)".
There is another common usage meaning "healthy" as in mame ni kurasu "to live healthy".
You may see it written as or .
The word is not new; it's been around for many hundreds of years, so I would not say that it is slang.
Links: Daijirin, Daijisen.