A common scenario in which you might need evidence of what a website page used to contain would be when you are making a claim for misrepresentation or false advertisement.
In this scenario, let's say that you entered a query for some service into Google's search field, and that Google returned a set of results for various providers of that service. When you clicked on one of those results, you were taken to (what is called in digital marketing) a "landing page" containing certain claims about the service offered by that provider. These claims might include things like price, features, and other deal terms.
Because your interactions with that website page (a.k.a. the landing page) were your only interactions with that service provider by that time, your entire understanding of the service provider's offering to you had to have been based on the representations made via that website page.
Because you deemed the deal terms represented to you on that website page to have been attractive enough for you to enter into that deal, you elected to enter into the deal by plugging your information into a series of form fields displayed that website page. Upon completion of that form submission, you had entered into the provider's advertised deal.
Sometime thereafter, you discovered that the deal you had entered into entailed hidden fees that were not disclosed on that website page. Specifically, one of the hidden fees purported to be for one of the line-item features that website page had represented to be included in the deal at no extra charge. So, you realize that the website page had misrepresented the deal to you, and you also realize that, given the fact that a website page can also constitute an advertisement, the falsity contained in that advertisement may also give you a cause of action for false advertisement.
But you need the evidence, and the single most important piece is that website page. So, you try to find the URL for that website page in your browser history, and...alas, you do! But when you browse to that URL, you discover that the content of the website page had since changed, and that some of the key misrepresentations on which you had relied before were no longer contained on that page.
You need to find the old version of that web page.
How to find old versions of web pages
One of the most effective tools for this purpose is called the Wayback Machine. Created and maintained by nonprofit library Internet Archive, the Wayback Machine is intended to be "a digital archive of the World Wide Web."
To that end, thing of the Wayback Machine as consisting of three key parts. First, there is a "crawler" whose job is to...well, crawl the internet, taking snapshots of millions website pages along the way. Second, there is an archive that stores all of those snapshots, each of which show a given website page as they were on a given date and time. Third, there is a web app (that is free to use) where you can enter in a given URL to pull up any snapshots of that URL the Wayback Machine had previously taken and stored.
Are archived versions of websites admissible as evidence in court?
The rules of evidence vary based on the type of proceeding and its venue. But, generally, speaking, yes, courts are open to considering archived versions of websites as evidence, so long as the court is convinced that what the evidence purports to be is authentic. (Example case: see Atl. Fabrication & Coatings, Inc. v. ISM/Mestek, in which one litigant introduced an archived version of a web-based terms of service.)