Your method maybe easier than what I am about to tell you but for some reason I can remember this URL.

1. Go to their channel
2. Select "Playlist" on their channel (the tob tabs) you should see the url end (or near the end) with a long list of alphanumeric characters
3. Paste this right before that set of characters http://www.youtube.com/playlist?list=PL
4. This will bring you to the playlist page. You can now select play all or play any of the videos

For example on Skillrex channel we have this playlist

More Monsters & Sprites EP

http://www.youtube.com/user/TheOfficialSkrillex#p/c/1B5CF096FA99B653

and switching out for the PL url

http://youtube.com/playlist?list=PL1B5CF096FA99B653

For the old (One Channel) design:

On the playlist page, click the "+ Save" button to save the playlist to your library as-is. A playlist saved like this will get updated as the original creator updates it. You can also clone the playlist by clicking the three dots > "Add all to..." button and adding all videos to a playlist you fully control.

The old design is currently available by appending &disable_polymer=1 to the URL, or by using Internet Explorer. It may disappear in he future though.

For the new (Material) design

On the watch page, you can save the playlist by clicking the 3-line icon with the plus next to it.

On the playlist page, you can save the playlist by clicking the very same icon in a different position.

Note that these options merely save the playlist to your library, they don't clone it for you to edit it. To clone playlists, you'll have to find a bookmarklet or an API program that basically iterates through all items of a playlist given playlist and creates a new playlist from your account. Creating a program like this is left as exercise to the reader out-of-scope for the Webapps Stackexchange and more something for StackOverflow. Though this exists if you trust an unknown third party application to take over parts of your account.