Literally: lion – child – vigorous – quick
Alternately: Highly-energetic, even violent activity. Doing things in a furious, intense, or overwhelming way. Ferocious, or at least energetic, like a lion.
Notes: This phrase apparently comes to us from the Mahaprajnaparamita Sutra (Japanese 『大般若波羅蜜多経』 = Daihannyaharamitta kyou, often shortened to 『大般若経』 = Daihannya kyou), and started its life in Japan as a Buddhist term, although it has since passed into broader lay usage.
While 奮進 (funshin) is a near-homonym and even a synonym to 奮迅 (they both refer to dashing forward energetically), replacing 迅 with 進 in this compound is an error.
獅子 is one of those words that almost demands you look up its etymology: why is there a “child” in lion? It turns out that the reason is simple phonetics: the Chinese adapted the Sankrit word for lions, simha (cf. the singa in Singapore) using 師 (shi) and 子 (zi), and then added the “beast radical” 犭 to produce 獅子. It helps that in Japanese, shishi can refer to “wild animals” in a generic sense, being both one of the readings of 鹿, “deer,” (cf. 鹿威し, shishi odoshi) and a component of 猪, inoshishi, “wild boar.”