There is no core Moodle tool that does record or playback, really don't need one. There is of course the PooDLL set of tools, but they are subscription based for v3.xx and while I expect it would be quite reasonable pricing, even that might be too much. There are a number of alternative that may already be available to you depending on your network set up.
If you are using the Adobe Suite, then Captivate might be available. Your system may use Camtasia, or a number of other proprietary tools. [These are the ones I know and have used, and I am not recommending anything here.] Have a good look, see what you already have. There are others, Jing, SnagIt and more I have never heard of I expect.
There is also a free, web-based, tool called ScreenToaster, Or if you prefer to have things done locally, there is Screencast-o-matic. This is downloadable, has versions for Windows or Mac, maybe Linux as well, can be used freely or if you pay a very small fee, you can get editing tools as well as the recording tools. Again, I don't know how good this app is, but at least two schools around me are using it for precisely the purpose you require.
The beauty of these tools is they can save recordings as MP4s and as long as they are not too long, they are not overly large files with enormous file sizes. You may have to restrict time length, it works out to about 10mb for 5 mins of video, well those I have tested have done so but please, don't take my word for it, it varies considerably and have yet to work out why.