Unfortunately a lot of times those splitters don't work with computer video cards. Computers and Monitors do what is called handshaking. This is where the monitor reports it's specifications to the video card. Unfortunately the Video cards are not generally built to be able to hand handshaking through a splitter so you often loose that signal. Because that signal is lost the computer doesn't think any displays are connected to it so it stops outputting video.
To really get the dual display like that i would recommend using an actual active splitter. This will help get around that problem. You could use one like this here:
DVI SplitterUnfortunately they're a lot more expensive than just the splitter. Another option is to purchase a new video card for your computer that has 2 DVI ports. That usually isn't as expensive as one of these splitters.