If a website only works while downloaded locally, you'll need to download it from the WayBack Machine to view it.
To download a website manually, follow these instructions:
1 - Click the "MD5 Checksums" button on the page for an archived website you wish to view.
2 - Go to each URL in this list, adding web.archive.org/web/0id_/ at the beginning of each URL (if the link is dead, try web.archive.org/web/*/ to see archives from other points in time).
3 - Save each URL in the same file structure.
Example:
The list of MD5 Checksums shows "7169936270B03B661027356F0D2F94BD *professorlayton.nintendo.com\unwoundfuture\_ui\img\flash\demos\hat.jpg",
so you go to web.archive.org/web/0id_/professorlayton.nintendo.com/unwoundfuture/_ui/img/flash/demos/hat.jpg.
You create a folder named unwoundfuture, and save the file in unwoundfuture\_ui\img\flash\demos with the same name.
You should now have unwoundfuture\_ui\img\flash\demos\hat.jpg.
Once you do this with every URL, the website should work offline.