How do you save data that is in fields inserted via hook_form_alter?
Other modules aren't generally programmed to be aware of your custom additions and don't know where/how you want to save the data. If you add fields via hook_form_alter() then you should also add your own submit and possibly validation handlers to the form similarly to this:

$form['#validate'][] = 'mymodule_someform_custom_validation';
$form['#submit'][] = 'mymodule_someform_custom_submission';


In the validation handler you can then validate your custom field data (if needed) and in the submit function you should save it. Small amounts of information can be stored in a variable, but if there's bigger amounts of data from different users you'll probably want to create a custom database table where to store all the information in.