hook_form_form_id_alter doesn't save data by default I use hook_form_form_id_alter() to edit the user_form. I'm creating some checkboxes and textfields. Thats working fine. I also can output the values, when clicking the submit button.

But it doesn't save the values by default. 

How do I save my values?
hook_form_form_id_alter only alters the form object for presentation / behavioral changes on existing forms. It doesn't alter processing or storage for existing forms. 

For that you'll need to add a custom submission handler, or use the Form UI to create new custom user fields if you don't need custom processing/storage needs.