This is a very useful feature, and I had the same confusion. The manual is unclear, really.
Hypermemory - is not another set of memories, it's better thought of as a bookmark, since it actually POINTS to memories (setup in banks). You kind of have to use Memory Banks to make Hypermemories useful.
The banks, for instance aren't memories either, as each memory can show up in one or more banks.
On an AM/FM broadcast radio, the buttons switch one or two settings (Set AM or FM + a station frequency).
Hypermemories set the entire radio's current settings: which bank is current, which Home frequencies, which band-edge memeories, etc. all at once.
I highly recommend the use of PC software to manage the memories of these (FT-7800, 8800, 8900). There are too many to keep up with and it helps you visualize and more easily setup the banks and hypermemories. I use G4HFQ's software and he has a version for the 7800: http://www.g4hfq.co.uk/ftb7800.htm
As to HOW to use them - if you travel, you can setup Hyper1 for City 1's repeaters, Hyper2 for City 2, etc.
I have an FT-8800 and I use one for local VHF/UHF repeaters, another for APRS + VHF repeaters, one for simplex (VHF/UHF), and one for 'drive-time', where one side of the radio is set to VHF repeaters, the other to just scan traffic reporter frequencies (in DFW area these are just above 450mhz).