Tonido.
I've only read the first 6 pages of this thread (then searched), but couldn't understand the multiple suggestions to buy Wifi Explorer Pro.
Tonido is free, and while it has the opposite client-server approach, for me, that approach has been much more useful.
You access files on your home PC from your Fire/phone. You get a permanent URL for the PC. It works anywhere on the internet. Upload, download, stream music, share files, browse photos, PDFs, Office files.
For example, I once went to lunch with friends, and needed information that was in a file on my home PC (which was running). In under a minute I had the Word file opened on my phone.
To do the ES Explorer sideload thing, you don't even need to touch your PC. Instead, upload the .apk from your phone to your PC, then go to the Fire and download the .apk from the PC.
You can install the PC server software in as many PCs as you want (I've got it in two, and thus have two different URLs).
One caveat, whenever I open Tonido on the phone/Fire, it pre-populates my last used URL and login/password credentials (maybe there was an initial setting where I selected this? But I can't find any setting to change it). So if you lose your Fire/phone, you better have Lookout or some other app to lock your phone/Fire remotely. Or maybe on the Tonido website there's a way to disable your URL(s). That "feature" is quite convenient, though, as it allows you to be within a few seconds of seeing a Windows Explorer view of your PC.
Tonido also offers to sell a NAS-like, low-power device, Tonido Plug, which you plug into your router. You can then access files on it remotely, without your PC needing to be on. I don't have one, so can't vouch for it.
What's a USB connection?