Unless someone can indicate otherwise, it seems AppBrain won't work with the eDGe.
On
www.appbrain.com, at the top right, is a Welcome message with a link to the "AppBrain app". When I click it, I get a page with no download link, only a "Download directly from the Android Market". Android Market links start with "market://", which won't work on the eDGe.
That same page says "Browse the app market on
www.appbrain.com from a *desktop* computer and use this app to install and uninstall apps with a sync". Note the requirement for a desktop PC. As far as I know, the eDGe doesn't have a sync-to-desktop facility, even if you're using a USB cable, or over wifi.
Similarly, when I investigated the "Fast Web Install" option, it says "Download Fast Web Install from the Android Market", which again you cannot do with the eDGe.
Individual app pages also don't have a download link (ultimately, you want an http:// link that ends with ".apk").
So, even though AppBrain is a third party market, it ultimately is still tied to the Android Market.
This leaves you two options: Google for things like "Pingchat APK" and take your chances, or download the apps to an Android phone and then copy the APK to your eDGe and install.