Every time you remove an app, it will remove its settings and contents with it. In case all of the above options fail, you can always remove the app entirely and then re-install it from the App Store free of charge. If no such option is there, try swiping a song from right-to left and see, if a red 'Delete' option reveals itself.

Check if you have an 'Edit' option or similar when displaying the songs inside 'Share It!' and once in that mode, see if you can select and remove songs. A lot of apps capable of receiving or creating user content have options to remove such items from within the app itself. You can also check, if you can remove the songs from within the app.

the songs should then be on the right-hand side and you can either simply remove them (hitting Backspace (Mac) or Delete (Windows) should take care of that) or export/import them to your computer.

check if 'Share It!' is on the left hand side and if it is, click it.

on the right-hand side (where you see the iPhone screens with your apps placed), scroll down (there is a file sharing section).

click on the device inside iTunes and navigate to the Apps section.

However, as with many applications, especially those promoting sharing, you may find an option to add or remove contents from within iTunes. This is an application-specific question regarding the app 'Share It!', which I do not know myself.

Shareit apple