Jo,

You aren't looking in the right box. At the Top of the LimeWire console (under file, view, navigation...) you see most likely 3 tabs to select. Search, Monitor, and Library.

You must be looking in the Monitor Tab, and have 'Show Incoming Searches' check marked. The listing directly below that are search terms that other users are searching for. I'm not totally sure, but I think whats happening in the network sends those terms to your computer and sees if it matches any in your Library. If you’re connected as an Ultrapeer these will fly by so fast you could hardly read them. As a peer connection, you get that Notice after you mark the check box saying, "Leaf node shielded by an ultrapeer..." Nothing to worry about just the way the Program works.
The songs on the bottom of the same screen, under the separating bar are songs that you have in your library that others are attempting to UPLOAD from your computer to theirs. NOT the Files you are trying to DOWNLOAD from the network.

SO.... When you type in your search terms, in the Search tab, the files listed there are the files that you can choose from to DownLoad to your Computer.... Just don’t change tabs when you search... Just stay in the window you typed your search in. There really is nothing in the others you truly NEED to know, its only background stuff that isn’t necessary to find music.