I don't know anything about CoolEdit but Audacity is a freeware program. The easiest way I can tell you to remove background audio is to:
1. Download and run Audacity.
2. Import your audio clip into Audacity. Make sure your audio clip contains at least a few seconds of uninterrupted background audio.
3. Select the seconds of background audio only.
4. Click "Effect" and select "Noise Removal / Get Noise Profile"
5. Select the whole file and Click "Effect" and select "Noise Removal" again. Only this time move the scroll bar to select the amount of background you wish to remove.
NOTE: the more you remove, the more it will effect your "Voice" Too little and you'll still hear music, too much and the voice will be distorted.
6. Click preview to analyze how your file will sound, BEFORE you click "Remove Noise"
7. "Remove Noise" and listen to your file. You can now select individual areas of the sentence. I.E. between words and click "Generate / Silence" to further remove background noise, as well as "cut" away unwanted bits, like the few seconds of background audio, used to get the noise profile.
8. Click "File / Export as WAV" OR "File / Export as MP3"
If your voice file still sounds strange to you, you can edit it with more of the features, changing speed, pitch, bass, etc until it sound like you want. BUT I suggest you save it AFTER you get the back ground removed first, so if it gets worse, you can start over from that point.