[logging] How do I get logs/details of ansible-playbook module executions?

The playbook script task will generate stdout just like the non-playbook command, it just needs to be saved to a variable using register. Once we've got that, the debug module can print to the playbook output stream.

tasks:
- name: Hello yourself
  script: test.sh
  register: hello

- name: Debug hello
  debug: var=hello

- name: Debug hello.stdout as part of a string
  debug: "msg=The script's stdout was `{{ hello.stdout }}`."

Output should look something like this:

TASK: [Hello yourself] ******************************************************** 
changed: [MyTestHost]

TASK: [Debug hello] *********************************************************** 
ok: [MyTestHost] => {
    "hello": {
        "changed": true, 
        "invocation": {
            "module_args": "test.sh", 
            "module_name": "script"
        }, 
        "rc": 0, 
        "stderr": "", 
        "stdout": "Hello World\r\n", 
        "stdout_lines": [
            "Hello World"
        ]
    }
}

TASK: [Debug hello.stdout as part of a string] ******************************** 
ok: [MyTestHost] => {
    "msg": "The script's stdout was `Hello World\r\n`."
}