Basically, a zip file is a file that's been compressed using the "zip" format so that it takes up less space. A zip file can contain one or more other files within it. It sounds like you may have created a zip file that contains allot more than the two songs you wanted to deal with.
I think you need to get the two songs out of the zip file. I think if you right click on the zip file you will see an "extract" option or something like that. Then you can either upload the two individually or create a new zip file with just the two songs.

You don't have to worry about anyone having access to your computer, but, if you upload that zip file you mentioned you will be uploading all of your files for others to see.