The concept of podcasts is made popular through Apples iPods and free iTunes application (mac/pc) but you can also enjoy them without (using a regular media player). Basically a podcast is just an RSS newsfeed with a few iTunes specific tags and a reference to a media file - in this case an MPEG4 video (encoded using the H.264 codec). When you "subscribe" to a podcast an application like iTunes will automatically download the media file and transfer it to your (video) iPod so you can enjoy the media during offline times, like when travelling.
In the top of the podcast pages you'll find links to an RSS feed and iTunes subscription and there are direct links to the video files on this page as well.
If the "Subscribe with iTunes" link above doesn't work for you, try this in iTunes:
Go to "Advanced > Subscribe to podcast..." in iTunes and enter the RSS feed URL. In the list of podcasts you will see all episodes from the RSS feed and you can download those you like to watch.
Enter the RSS feed URL in an RSS newsreader. When new episodes are published you will see them in your newsreader. From there you will find a link to the media file. Download the file. Should play in QuickTime on Mac/Windows and VLC on Linux.
The size of the podcasts are typically between 50 and 150 MB depending on length and content. The typical length is between 10 and 30 minutes.