What does DNS flushing do?

DNS is the Domain Name System. DNS servers convert a domain name (such as example.com) into an IP address (in this case 192.0.32.10). The mapping of names to numbers can change from time to time.

Your computer holds a record of DNS entries to save looking them up every time. This is your DNS cache. You can delete those records (flush the cache) any time you like.

If a website has recently moved servers, you might see the old website for a while. Flushing your DNS cache might help.

Whenever you type a URL on the adress bar it fetches the corresponding IP address to communicate with web server (it might be any kind of server). When you repeatedly use a same URL its just a waste of network resource to fetch corresponding IP adress everytime because IP dont change too often.

so your computer stores the combinations of domain name and its IP in local cache to avoid fetching from Domain name Server(DNS) everytime you use same domain name(URL).

It also holds another crucial info called "Timeout" which says about the valid time for the IP and domain name combination, when this time times out your computer re-fetches the combination from DNS and stores in local cache again.

Dns flushing is the mechanism where the user can manually make all the entries in the cache invalid, so your computer re-fetches new combinations by now on whenever it needs and stores in local cache.

Your system holds the IP address of named sites... When you do a ipconfig /flushdns, your system clears the cache of name to ip entries and reloads them from the connected DNS server.

Unless you're on old versions of Windows, it means that the website operators messed up. They moved the service to a new IP address; before doing that, they should have lowered some timeouts on the name data in DNS, so that they said something like "this mapping is valid for 5 minutes" instead of "this mapping is valid for the next day". This needs to be done some time in advance.

The exception to this is that old Windows releases (through XP? I forget) would ignore timeouts shorter than a day and just remember the old mapping anyway. While it's common for caches to have a "lower bound" on how long the timeout can be, it's usually around "5 minutes", not "24 hours". This means that a website moving IP address has to actually be available on two different IP addresses for that length of time.

The solution for the website operators is to have a forwarding "proxy" run on the old IP address, passing traffic to the new IP address, for a couple of days, to give time to deal with the sad reality of stupidly excessive caching.

Either way, the problem has been punted onto you. But there's a reason that big professional sites don't ask you to do this -- they work around the problem themselves.