What is an Adaptive Mean Filter?
A wild guess. $L$ denotes a region $R$ around pixel $g(x,y)$, potentially a symmetric one of size $(2L+1)\times (2L+1)$. $m_L$ and  $\sigma_L^2$ are the average and variance in $R$. And $\sigma_n^2$? Probably the estimated noise variance over the whole image, supposed stationary.

In a flat region with noise $\sigma_L^2 \sim \sigma_n^2$, the result could be close to the average. Around edges, one expect an higher response, so sharper. Potential problems arise when  $\sigma_L \ll \sigma_n $. Another interpretation: 

$$\hat{f}(x,y)=(1-\frac{\sigma_n^2}{\sigma_L^2})g(x,y)+ \frac{\sigma_n^2}{\sigma_L^2}m_L(x,y)$$
is a sort of weighted mean (since weights sum to one) of the image and its smoothed version, whose dependency on $(x,y)$ is made more explicit.