How do I find the hooks used by this module? How can I find out what hooks are available in the Entity Clone module? I want to modify the entity after it is cloned to change its author and workflow state.
As @Kevin says, modules should have a modulename.api.php file with examples that implement hooks that the module provides.

When this file is not available you can search for the code that creates a hook in Drupal.

Drupal 8

Look for following functions of the ModuleHandlerInterface, usually implemented by the ModuleHandler class:


invokeAll
alter


Using grep it's easy; just go to the module's root directory and execute:

 grep '\-&gt;invokeAll(\|\-&gt;alter(' * -r


This command looks for the strings -&gt;invokeAll (creates regular hooks) and -&gt;alter (creates alter hooks).

For example, if you use this grep command with core's module User you get:

src/Entity/User.php:    \Drupal::moduleHandler()-&gt;alter('user_format_name', $name, $this);
user.module:  \Drupal::moduleHandler()-&gt;invokeAll('user_login', array($account));
user.module:    \Drupal::moduleHandler()-&gt;invokeAll('user_cancel', array($edit, $account, $method));
user.module:  \Drupal::moduleHandler()-&gt;alter('user_cancel_methods', $methods);
user.module:  \Drupal::moduleHandler()-&gt;invokeAll('user_logout', array($user));


So, this module creates the following hooks:


hook_user_format_name_alter
hook_user_login
hook_user_logout
hook_user_cancel
hook_user_cancel_methods_alter


Drupal 7

Same procedure, but in this case you have to look for the functions module_invoke_all() and drupal_alter().

grep 'module_invoke_all\|drupal_alter' * -r


Keep in mind that this method is not 100% reliable, as some modules use their own hook creation implementation (I've seen this in Drupal 7 mainly), but it's a good starting point if the said modulename.api.php file is missing.

Entity Clone

Unfortunately Entity Clone doesn't create any hook, and the command yields no results.

invoke

There are certain hooks that are only invoked in a particular module, see ModuleHandler::invoke and module_invoke. As they only affects certain modules they are less used than regular ones (they are some kind of internal use hooks).