Please use the Question Tags to state the name and full version number of the Story Format you are using, as answers can vary based on that information. Based on your reply to HiEv I will assume you are using Harlowe v3.0.2
I had a quick look at the source code of the HTML page you linked to, it is using links (and JavaScript) to either:
a. hiding sections.of existing content while reveal other related sections.
eg. the replacing of the "shabby muslin dress" link with the "long crinoline gown" link, and vice-versa.
b. replace marked sections of text on the page with alternative text.
eg. the "with disdain" / "with admiration" text switching when the above mentioned two links are selected.
You could use a Hidden Hook combined with the (show:) macro to do the "reveal hidden text" effect.
The (link: "quick")[very quick(show: ?dogcolour)] brown fox jumps over the lazy |dogcolour)[black ]dog.
You could use a Named Hook combined with the (replace:) macro to do the "replace marked section of text" effect.
The quick brown fox (link: "jumps")[jumps(replace: ?lazydog)[disinterested cat]] over the |lazydog>[lazy dog].
There are also the (link-reveal:) macro, the (link-show:) macro, and a number of (click:) related macros to could use.
However as stated by HiEv crafting a Passage to work exactly like the one you linked would take quite a bit of work and would likely require the usage of displaying the content of 'child' passages within the current one (to make the code required to do what you want more readable). And like HiEv I would suggest starting with some smaller/simpler prototying projects/passages to learn how to do each of the effect types to require.