If you wish to connect to one of these servers and enjoy it for yourself, you can use the connection information on the detailed page of each server. This is in the format of hostname.com ####, and if you have a Telnet client you can connect directly to that hostname and port number. If you don't know what Telnet is, you shouldn't use a Telnet client. In fact, Telnet is a very basic way of interacting, and there are a number of specialized clients to connect to these servers. Here are a few, but by no means all. Some servers, such as BatMUD, even maintain their own clients.
Website: http://www.gammon.com.au/mushclient/mushclient.htm
Platforms: Windows
One of the more advanced clients on the market, MUSHClient has many of the expected features one wants, like scripting, triggers, aliasing, and 256 colors. It also provides a number of advanced graphical and automapping features, and supports clickable links and other advanced MUD features via the Mud Extension Protocol(MXP).
Website: https://beipdev.github.io/BeipMU
Platforms: Windows
BeipMu is a growing popularity client which is a bit more lightweight than MUSHClient, but has grown to support many of the same advanced features.
Website: https://www.mudlet.org/
Platforms: Windows, Linux, MacOS X
Another advanced client, which includes automapping and so on. This one is cross-platform, though occasionally encounters compatibility issues with some MUDs. These occurences seem to be happening less often as the developers have been consistently improving the client.
Website: https://www.potatomushclient.com/
Platforms: Windows, Linux, MacOS X
A popular cross-platform client, this isn't quite as feature-rich as the ones above, but has an easy to use interface and especially friendly means of spawning new windows to organize chat.
Website: https://riverdark.net/atlantis/
Platforms: MacOS X
The only fully Cocoa and Mac OS native client, Atlantis lags behind in some of the more advanced features, and as of this writing doesn't have good scripting support, but has a much better integration with the Mac interface.
Website: https://tinyfugue.sourceforge.net/
Platforms: Windows, Linux, MacOS X
The grandfather of all these clients, TinyFugue has complex and powerful scripting tools and automation, but lacks any of the fancy graphical enhancements and is somewhat limited in what it can display due to its reliance on the terminal. This runs in a terminal window, so it is ideal for remote sessions, but falls short on user-friendliness. It is often included in Linux repositories, under tf for version 4, or tf5 for the newer version 5.
Website: https://github.com/ingwarsw/tinyfugue
Platforms: Windows, Linux, MacOS X
A more recent expansion on TinyFugue, this adds a number of features long-requested such as colorized logging and support for more modern scripting languages like python and lua.
Website: https://bt.happygoatstudios.com/
Platforms: Android
Basically the only game in town for Android-based clients, Blowtorch has a number of the expected scripting and options, but far fewer than more full-grown desktop clients. It also has the various problems all mobile clients have with disconnects while sleeping, and a limited number of windows open at one time.
Website: https://github.com/splinesoft/MUDRammer
Platforms: iOS
Just as Blowtorch is the only reasonable client for Android, MUDRammer is basically king on iOS. None of the admin here use it, but anecdotally it is somewhat superior to Blowtorch, but still inferior to the desktop clients due to the limitations of the mobile device format.