What is Local Mean Filter?
They probably just wanted to say that the image was blurred (by some local method, e.g. convolution with a Gaussian kernel) in a more scientific way. 

On the sharpening: They don't sharpen the image directly. What they do is to blur the image and the subtract the blurred version from the original. The result is the same as sharpening directly via an appropriate filter. 

Why this works

Assume the image $I$ and an blurring kernel $g$ (which could be a Gaussian kernel). In Fourier-Space, blurring the image by convolution becomes a multiplication:

$\mathscr{F}(g\star I) = \mathscr{F}(g)\mathscr{F}(I)$

with $\mathscr{F}(\cdot)$ being the Fourier-Transformation. 

Then $I - g \star I$ becomes: $\mathscr{F}(I) - \mathscr{F}(g)\mathscr{F}(I) = \mathscr{F}(I)(1-\mathscr{F}(g))$.

Since $\mathscr{F}(g)$ enhances the low-frequency components in in Fourier-Space, $1-\mathscr{F}(g)$ enhances the higher frequencies (I assume here that $\mathscr{F}(g)$ is normalized appropriately in magnitude, which can be achieved via scaling factors). So, in image-space, the resulting image is a sharpened version of $I$.