First, you'll want to clean the screen. Carefully clean it with a lens cloth and lens cleaning solution, either purchased from the store or mixed yourself as 50/50 isopropyl alcohol and water. Don't put too much pressure on the screen while doing so, as that could cause permanent damage—restrain yourself to small, circular movements.
Once you've got this done, it's time to make the screen protector. You'll want the lens protector to be a close fit, so it's best to trace out the outline of the LCD screen first. This probably easiest to do on the back of the plastic itself (before it's been removed from the backing.) If you don't feel comfortable tracing the outline through something opaque, then you could use translucent tracing paper and then transfer the outline, or you could measure the dimensions and then draw that out onto the backing. Whatever works best for you—just make sure that this is as accurate as possible.
Tracing slightly bigger is better, but be careful that the shape is not so large that it will stick out over any edges. This will present possible points from which the plastic could start peeling away from the camera. Fit it such that
Now, placing it onto the screen. Start with a small edge, and gradually put it on, pressing out any air bubbles that form between the LCD screen and the protector as you place it on. Once you're done, press down the edges so that none of them curl up—again, this is where the plastic will start peeling away, and this is to be avoided. If there are any places where it sticks out, try to trim it with the scissors—if that proves too big, then try nail trimmers or the like. (But don't stab the screen!)
Once the plastic is down to your satisfaction, you're done!