In case you had a successful inbound call where your user spoke to the caller, you would be able to play your call recording in your inbox (within a few minutes of the call).
Sometimes you may notice a recording that says "This call recording does not exist....". This happens because you may have not checked the Record this call? option in your custom app. To change it -
1) Go to App Bazar
2) Go to the Installed Apps page
3) Click on the 'Edit call app' link of the required app (which handles calls to your Exotel number)
4) Navigate to the Connect applet
5) Check the 'Record this call?' option
If you haven't enabled this option the calls will not be recorded.