What purpose does it serve for an administrator to access a remote
desktop in  a network?

If you have a big network, it's practical to use a
remote-desktop-software, because you hadn't to go around if you want
to do something on a computer in the network. For example, if you have
some datas on one computer but are sitting on another and you need
them, you can log in via remote-desktop and get the datas.

With a remote connection to a desktop, you can Increase respons times,
Example if you are the administrator and you are at home and have to
drive for 45 minutes to get to work, you just saved 45 minutes of
downtime, by walking to your computer at home and remotely accessing
the server to resolve the issue