on client side, you should use the XMLHttpRequest object to connect to the PHP page, download the schedule data, and then use it to create the widget's mini-view.
On server side, you PHP script should retrieve this data from your MySQL server and then send it to the client by using a suitable data format: usually, JSON or XML.
If the schedules are mantained into separate HTML files, you could simply redirect to these pages by using a PHP code similar to this:
Anyway, using HTML files as data format can have some drawbacks if you need to manipulate this data also on client side.
Hope it helps,