The column to which you refer is the Name column, which shows "filename.extension". The extension is "contact". I tried just now to not show that column and was unable to hide it. However, you can make it narrower and relocate it off screen to the right so it's effectively out of your way. In its stead, you will then want to show the "Full name" column so you will know to whom you are sending mail.
I make a habit of selecting the "Last name, First name" version for the Full Name field when creating or editing a contact. Do that by pulling down the arrow on that row on the "Name and E-Mail" tab. The name of the file will then display that name in the Contacts listings, as will the "Full name" column.