iExplorer is a transer utility for iPhones, iPads, and iPods, and in order to transfer the data from your device to your computer, the data does need to be physically stored on your device’s hard drive, so this is why visual voicemail is required (for voicemails only). If you don’t have visual voicemail, then the audio files are not actually stored on your hard drive, and they are only stored on your cell provider’s online servers where they make you call in and listen to them via a phone call. Our Support Center article below explains more:
Why are my voicemails not appearing?
The article above also includes a link to an Apple article that lists each and every cell phone service company and says whether visual voicemail is possible or not. As you mention, in some cases, a cell provider offers visual voicemail, but they may charge an extra fee for it. We obviously do not profit from that fee whatsoever, and most cell companies include visual voicemail with the basic rate for your plan.