Add the following intent filters to your Activity where you want the picking to take place:
<intent-filter >
<action android:name="android.intent.action.PICK" />
<category android:name="android.intent.category.DEFAULT" />
<data android:scheme="file" />
</intent-filter>
<intent-filter >
<action android:name="android.intent.action.GET_CONTENT" />
<category android:name="android.intent.category.OPENABLE" />
<category android:name="android.intent.category.DEFAULT" />
<data android:mimeType="*/*" />
</intent-filter>
The first one handles an Action Pick, and the second one Get Content.
You may want to change your mimeType to restrict selection a little. The one I provided will put your app in the selector for every type of file.