The message will be stored in the base unit which, unlike the handset (which has a jack socket for use with a headset) doesn't have any sockets available for outputting sound. So (although it's not ideal) you'll have no alternative other than to place a microphone next to the speaker in order to make your recording.
While you could get hold of a microphone from somewhere (making sure that it's got a 3.5mm plug on the end or that you've got an adapter so that it can be used with a 3.5mm socket), plug it into your computer and then record the sound input, it would seem far simpler to use a microphone and a recording device that you've already got. Look no further than your mobile phone!
Almost all mobile phones have got a 'Memo' function on them, which allows you to record sound input from the phone's microphone. Find it and, if you've not used it before, experiment with it a bit so that you know how to get decent recordings. Then play the answerphone message and record it onto your mobile phone.
Sound recordings can be transferred from your mobile phone to your computer (and then to back-up media, such as USB memory sticks) in exactly the same way that photographs can. i.e. connect your phone to your computer with a USB cable, look for it in Windows Explorer as an external drive, locate the relevant folder and then drag-&-drop (or copy-&-paste) the relevant files from one device to the other.