Meaning of mame
As Dono explains, this can mean "diligent; serious; hardworking; hale, healthy".
Etymology of mame
Reading
Although this has the same kana reading as 豆{まめ} "bean", it isn't the same word. "Bean" is pronounced with a high second mora and then an immediate drop, as in まめは【LHL】, while "serious" is pronounced with the so-called heiban or "flat" pitch accent, where the pitch rises on the second mora and then gradually comes down, as in まめに【LHH】.
Derivation
Derivationally, Shogakukan states that mame "serious" is probably an alteration from 真{ま} ("real, true") + 実{み} ("body, content; fruit, results"), or from 真{ま} ("real, true") + 目{め} ("eye; appearance"). Meanwhile, the entry at nihonjiten.com suggests a shift from 真{ま} ("real, true") + 実{み} ("body, content; fruit, results"), an abbreviation of 真面目{まじめ} ("serious"), or a shift from 正{ま} (irregular reading, "real, true") + 身 ("body, content"). Dropping out the じ from the middle of まじめ seems unlikely to me, so the other theories are probably closer to the root of the matter.
As Dono notes, まめ in this sense has been in use for a long time. Its earliest appearances are probably in the Nihon Shoki, finished some time around 720 CE. In section 11 of volume 14 about Emperor Yūryaku, we find the following text:
This is in 漢文{かんぶん}, which is essentially Classical Chinese, so I'm unsure of the meaning, but it seems to be "seriously shining sun" or something to that effect. At any rate, this まめ does appear to be the same word, indicating a very long history in the language. As Dono notes, this is not slang.