Firstly, your google searches might yield a few more results if you specify what you mean by memory. It is an ambiguous term because it can refer to harddrive space, or ram, or rom or something else. I am going to assume you mean system memory
Secondly, you should work out what that button actually does. Something like this might shed some light: http://android.nextapp.com/site/systempanel/doc/taskmanagement
Thirdly, because the memory used by the OS and other applications cannot be controlled directly from your application, any memory clearing call will have to be a system call to the OS. So there should be some kind of API you can use to interact with the system and ask it to control other apps.
Android-Close Other Apps This link shows how to close other apps (maybe a good starting point)
This class interacts with other activities on the system:
http://developer.android.com/reference/android/app/ActivityManager.html
Either way it is a tricky thing to do as your first app. Goodluck!