Capturing streaming audio from a radio transmission means that you are telling your computer to grab the digital audio signal and save it as a file that you can listen to. As commonly practised by radio stations nowadays, the BBC releases many of its broadcasts and online content through its iPlayer software, which treats the streams just like any other. You can use a free software called Audacity to capture these streams.
Go to the Audacity website and click on the corresponding download button for whichever system you are using (Windows, Mac OS X, or Linux).
Click on the EXE file once the installer has downloaded, then follow the Wizard's instructions for installing the software.
Double-click on the Audacity icon (on your desktop or in "Start") and then double-click "Programs".
Click on "View" and then "Float Mixer Toolbar" to enable you to choose the audio input source, either "Wav Out" or "Stereo Mix". This choice depends on whether you want the sound to be recorded onto your computer in stereo or mono.
Open the BBC audio streaming window by clicking on its link in whatever Internet browser you are using.
Click on the "Record" button in the Audacity program window while the audio window is buffering the stream. When the stream finishes, click on the "Stop" button to end the recording.
Click "Save" to save the stream as a WAV or MP3 file.