Since the DMZ is lower security level than inside, you must create and apply and access-list to allow DMZ-originated traffic to access inside addresses.
Something like:
access-list DMZ_IN extended permit
access-group DMZ_IN in interface DMZ
Inside to DMZ will automatically work (unless you start ACLing in in which case an implicit deny will be added at the end).
If you're new to the ASA, I recommend you use ASDM to create your changes. Set it to preview commands and look at what it generates to understand the CLI.
