Without knowing the brand of MP3 player, I don't know exactly how you'll get the files onto the device. You'll need to follow the instructions that came with it, or any software that was included.
It may just appear in the Finder as an external volume. You could then drag and drop the .mp3 file onto it. Or try software like https://www.doubletwist.com/desktop/
Either way, you have a couple of choices to download the files.
You can use iTunes to download BBC Radio content. Just open iTunes, Select the iTunes Store on the left and search for the content you want. Once you've selected and downloaded, it will appear in Podcasts on the left.
If your MP3 player is an Apple device, you can sync to it directly from iTunes Otherwise, right click on your Podcast and choose Show in Finder. This will reveal the MP3 file in the Finder and then copy it to your device (see above).
Or,
You can download MP3s directly from the BBC at http://www.bbc.co.uk/podcasts
Right click and select, Download Linked File. The file will then be downloaded and you can move it to your device.