How to save custom fields when using hook_form_FORM_ID_alter?
The only thing your form alter has done it added fields to a form. But you need somewhere to store the values of those fields.

When you add a field to an entity (node) using the Field UI module Drupal creates 2 database tables. One to store the field values and one to store older revisions of that fields value. Without these tables you don't have anywhere to store the values.

Here are steps to add that field your node bundle:


Make sure the Field UI module is enabled.   
From the admin toolbar visit Structure > Content types > AYG (the name of your node type) > Manage fields.
On this page add your new fields and configure as required.


Now your new fields will show on your node creation / edit forms and you will have tables to store the values of those fields.