It's actually quite easy, but time consuming. It really depends on how they are doing the stream and if they are saving the recording for later viewing. If they are saving it for later there are a couple of things you can do.
The easiest thing you can do is use a program called Jdownloader. It's java based, so it runs on Windows, Mac, Linux, etc. Run the program, copy the link to the stream you want to save, and paste that into Jdownloader. It will then download whatever is in that link.
If that doesn't work, what you do is visit the stream, if using Chrome, go to View->Developer->Developer Tools. Click Network in the developer tools section at the top. Start/Play the stream, and then pause it. In the Developer Tools window, scroll back up to the top, and you're looking for a name in the name section that includes ".m3u8" or something like that. Select that entry and if using Windows, right click and select Copy->Copy link address. If you using Mac, hold down control and do the same thing. Then paste that link into Jdownloader and download.
Jdownloader works with YouTube videos too, but you want make sure you select the best video quality before copying the YouTube link and then paste that into Jdownloader.
Hope that helps.