MobileMe members can access any file on a hard drive connected to either product via the internet. The older versions allowed hard drive sharing to people on the same network. Now the feature has been extended to allow you to get these files from anywhere.
You initiate the sharing by registering your Airport Extreme or Time Capsule with your MobileMe account. When you are away from your Mac, that hard drive will show up in the Finder sidebar. The drive must be a USB drive -- there is no support for either flavor of Firewire on the AE or TC.
It does not appear that this remote sharing feature has been added to the last generation of these products, although a firmware update might be able to accomplish it. No word from Apple on whether or not this will happen. Update: the Apple Sales internal site (thanks Mr. X) is listing the following... looks like that feature is going to work on the older gear!
Note for existing AirPort Extreme and Time Capsule customersThis update is obviously designed to boost the attractiveness of the MobileMe service, to which Apple has been showing increased attention of late. Apple specifies that Leopard is needed for this feature, so Windows subscribers to MobileMe appear to be out of luck
Existing AirPort Extreme (MA073 and MB053) and Time Capsule (MB276 and MB277) customers can access their AirPort Disk / Time Capsule disk through Mobile Me over the Internet by upgrading to AirPort Utility 5.4.1 and AirPort firmware 7.4.1. Subscription to Mobile Me online services is required for this feature. Simultaneous dual-band Wi-Fi and Guest Networking features are only available to new AirPort Extreme (MB763) and Time Capsule (MB764 and MB765) customers.
Thanks to everyone who sent this in.