The way you have things now, when you press the submit button, the page sends an HTML POST request to your PHP page. Then the PHP page loads in your browser.
Instead, you’ll need to use the jQuery post method to submit the information to the PHP page and retrieve information sent back: jQuery.post() – jQuery API
It’s a little complicated, so I’d advise to start out by doing a little googling for tutorials and examples as a first step, then come back with more specific questions as they come up.