What does WWW do?
www is a subdomain of example.com.
You can configure DNS to have multiple sub domains which resolve to different web servers and hence give you a different site.
example.com (no subdomain)
site2.example.com
site3.example.com
It sounds like whoever set up the network set example.com to resolve to the internal web server, on the networks DNS server; this is accessible only to hosts on the internal network, so anyone who browses to example.com inside the network sees a different website from those externally, as they use a different DNS server to resolve example.com. It may have been better to give this its own sub domain. e.g internal.example.com
A user inside the network browses to www.example.com hits a link and is taken to example.com which is fine externally, but internally they resolve to different web servers, which is why you have been asked to change the links.
If the content on the internal server is the same as the external, then you could end up with the reverse problem. When browsing the internal site, any hyperlink takes you to the external site.
If possible you might find it better to use relative URLs in your hyperlinks. e.g. <a href="/index.htm">Index Page</a> instead of <a href="https://example.com/index.htm">Index Page</a>
The browser would then choose https://www.example.com or https://example.com depending on the site that you browsed to.
Why the www works could possibly be Wildcard DNS entry, which means *.example.com resolves to the same IP, effectively a catch all for sub domains that don't exist.
WWW is an acornmyn for the World Wide Web, it also happens to be a common subdomain employed by people to designate that the domain is hosting a web site.
The use of a subdomain name is useful for load balancing incoming web traffic by creating a CNAME record that points to a cluster of web servers. Since, currently, only a subdomain can be cname'ed the same result cannot be achieved by using the bare domain root.
In most cases I believe the www. subdomain is omitted and it doesn't really matter if you type the one or the other. However, given what the web looks like today and seeing as how HTTP and the web is really ubiquitous, I would never mandate the inclusion of www. in the domain name and if you look a web sites today you'll notice that sites such as facebook and google prepend www., even if you don't type it.
Why it works even when you type other things has to do the way your DNS is setup for that domain. There are ways to handle wildcards and forward all traffic regardsless of subdomain to the same web server. But it's up to the web server to respond to each individual host header in some manner and this is entierly configurable (and obviously you need to check what the configuration is to answer the question in full).
If they are trying to use exactly this "url": http:\example.com\PAGE.XXX then nothing will help them.
As of www suffixes in www.example.com ...
Historically 'www' as 'ftp', 'pop3', etc. were names of machines running corresponding services inside 'example.com' domain. And actually this schema is still used in many companies.