'Podcast' used to have a specific meaning, referring to the method of delivery. These day podcast is usually synonymous with download.

The BBC seems to use 'podcast' to refer to downloads of radio programs for which it owns the copyright, and do not have to be played though iPlayer or their own website. They are usually (always?) part of a series, like In Our Time, rather than one-off programmes. They are usually available to download for a long time, and can be kept for ever.

If you go to the BBC programme page - the one with 'programmes' in the URL (which is accessible from the BBC Sounds page which the OP linked to) then you'll see a Download button. You get an mp3 or similar which you can play on any media player.

https://www.bbc.co.uk/programmes/m000qm0d

There are other ways to get BBC podcasts, for example from a podcast app which will automatically download the next one in the series. It depends what computer or mobile device you have.