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 mins of the call).

Sometimes you may notice a recording which says "This call recording does not exist....". This happens because you may have not checked the Record this call?  option in you custom app. To change it 

1) Go to the Installed Apps  page

2) Click on the 'Edit call app' link of the required app (which handles calls to your Exotel number)

3) Navigate to the Connect applet

4) Check the 'Record this call?' option