You can use the magnetic lasso tool (or regular lasso tool) to crudely select the logo.
Then, invert the selection (Selection>Invert), and hit the Del key.
Alternatively, you can select the background with the magic wand tool and delete it(I will talk about an effect to erase all the little dots later)
If you used the lasso tool, then use the magic wand tool to select the black background
and hit Del
You can optionally use the eraser tool close to the edge, but it's not really necessarily because the what I talk about next should get rid of the edge. If you want to make sure you dont erase the logo at all, use the magic wand tool to select the transparent area, and then use the regular lasso with Shift to add all the little dots. It should go fast if you use large circles to add all the dots.
To make jagged edges and little gray dots disappear
Whatever tool you used, you will probably need to remove the jagged edges and little gray dots it left behind.
You can select the transparent area with the magic wand tool and then select inverse
then create a new mask (below all your layers rectangle with a dot in the center)
then go to
filter->blur->gausian blur and adjust the value until your jagged edge is highlighted (around 2.0 is ussually enough. Make sure you have the mask selected and not your regular layer otherwise it wont work )
Click OK and then go to
image->adjustments->levels and play around with the sliders on the top (the one that has 3 sliders with the gray one in the middle)
Move the sliders beginning with the left one until the jagged edges are gone. Then move the the right one until the whole image is visible.
The jagged edges should now be gone and so should the background...
You can also use the eraser tool and make it really large with the
] key, then click on the edges, and then start using the magic wand tool .