How do I (in general) track down what happens on form submit? I am working on a site that uses the Chosen module. There is a field that I need to edit programatically. When I use the actual field edit interface, Chosen adds an option called "Apply Chosen to the select fields in this widget?" When I set that value and submit the form, all is well... it changes what I need changed. However, I need to be able to track down what this is actually doing on the back end so I can change the value of this option for this field programatically... once... in an update hook. The user needs to still be able to edit this value using the form... I just want to flip its value this one time. I don't know where to start looking. In the Chosen module, I can see where it adds the option to the form, but I don't know what piece of data that option is stored with, nor how to see what the form is doing when it saves it so I can duplicate that functionality. It's hard to even know the right terminology to ask this question.
You mean updating the field instance settings programmatically, in an update hook, hook_update_N(). Do it as follows:

function YOURMODULE_update_N(&amp;$sandbox) {
  // First load the field instance.
  $instance = field_info_instance($entity_type, $field_name, $bundle);
  // Make the changes in the field instance configuration.
  $instance['widget']['settings']['apply_chosen'] = 1;
  // Save the field instance.
  field_update_instance($instance);
}