In order to play Podcasts directly from your Browser, you MUST have an MP3 player extension installed for your browser.
Go to your browser setting page and select Apps or Extension and then search for an MP3 player and add it to your browser.
For Firefox Click Here and install the plug-in
For Chrome Click Here and install the plug-in
If you want to download the file, the MP3 player will give you a download link on the player screen when playing the file.