If I need it for any other page, do I just refer to it's class name?
No, $urls takes an array of URLs you would like to cache.
And I'm not understanding what the $original is doing?
$original is the previous saved state of the page this is useful for doing your own custom detection. For example depending on fields that have changed (eg $URLSegment is different), the pages that have been affected by that change could be different.
The example code has some array_merge in there
You would only need to do that if you needed to merge 2 arrays. That code you posted has another function subPagesToCache() which needs to have its values merged with the current function.