Have you tried reading the instructions linked to from the download page?
You just unzip it and run the executable in the top level directory. If you need more specific instructions you need to ask more specific questions.
When I download the zip file, I can only open it with the 7zip. I extract to a location. Then select sqldeveloper.exe and a cmd prompt box opens for a second and disappears and nothing else happens. I hope this is more detail.
Try opening a Cmd window and running ...\sqldeveloper\sqldeveloper\bin\sqldeveloper.exe. That way any error messages will go to the console window and it will stay open for you to read it.
Usually downloading the 32-bit SQL Developer with the 32-bit JDK included works fine on both 32-bit and 64-bit Windows. Just make sure you have write access to the \Users\<your_userid>\AppData\Roaming directory: that is where the user-related SQL Developer settings are typically stored (unless somehow customized) on Windows 7.
SQL Developer Team