Why can't I play my iTunes songs on my computer?

I'm not sure exactly what you mean. Do you mean that the iTunes application won't launch on your computer? Or it launches but you don't see any tracks in it? Or do you see the tracks but when you try to play them, you don't hear anything? We'll need more details before we can offer much in the way of useful suggestions.

Regards.

To expand on what varjak paw has told you, "iTunes" is the application. Music purchased from the iTunes Store is "music", not iTunes or iTunes music. (Call me pedantic, but it's an important distinction that helps avoid the sort of confusion we have now.)

Assuming that you do have iTunes (the application/programme) installed on your computer, then to get the music that you purchased from the iTunes Store to show up in your iTunes Library, you need to sign into your Apple account (the same account that you used to buy the music) from iTunes on the computer. You then go to your account page and download the purchases from there.