The way your image is displayed is misleading. It has an alpha channel. This is why the results from Binarize are weird.
Check this:
im = Import["http://i37.ltalk.ru/22/85/298522/92/9427492/leave2.png"]
ColorSeparate[im]
From the output it is clear that the alpha channel is already a good mask of the leaf.
To compute the area, use
comp = Binarize@AlphaChannel[im]
ComponentMeasurements[comp, "Area"]
(* {1 -> 82093.4} *)
As we're working with an image, the units are pixels.
To get the result in
$\mathrm{cm}^2$, we would need to know the dimensions of the image or the leaf in centimetres. If you have the actual leaf, you could measure it with a ruler, then use that information to calculate the pixel/cm ratio.