Apple - What's a good iPhone app that will let me view files on my network?

I think NetPortal by Stratospherix may do what you want. I use their FileBrowser for iPad, but I haven’t tried NetPortal, which is their iPhone version. They also have a free NetPortalLite version.

(Sorry, I don’t have the rep to include more than one hyperlink.)

You are going to need an AFP or SMB client then - thats what the machines themselves are doing. I don't think there are any AFP or SMB clients in the app store - therefore you may have to jailbreak your phone to use one - and after a few searches, still don't know if there are any available (I do however find a lot of options for mounting your iPhone on your Mac).