If I understand correctly ...
1. You want to load the page with the form
2. When someone selects an item in Box A
3. Box B is populated with values based on Box A's selected item
4. And not reload the page or submit the form at this time
You can not do this with PHP alone. You would need to submit the form and thus reload the page in order to just use PHP to do this.
1. Detect when a new item has been selected in Box A
3a. innerHTML method, or
3b. DOM methods
Hope this helps, let us know if you need more help with this