The Internet is Stateless
Anything done with JavaScript on the client side is not automatically remembered between page reloads due to the stateless nature of the internet. Submitting a form via post will result in a page reload so client-side changes are forgotten. To remember the changes made, the developer needs to specifically code a storage method.
Ajax
As you are using JavaScript and specifically jQuery already, an ajax call could be used to notify the server of the changes made client side.
jQuery has an excellent AJAX library. Several tutorials are avalailable at the jQuery tutorials page. There are also plenty of web pages devoted to the subject.
Ajax example using jQuery
As a simple example, when you add the extra row you could call an ajax function such as:
$.ajax({
url: 'ajax/add_row.html?user_id=100&data=new-data',
success: function(data) {
alert('Adding of the row was acknowledged.');
}
});
Server side
On the server side, you simply need a page located at ajax/add_row.html to do a little bit of work, taking the data passed to it and adding it to the database. On the next page reload, the added data can be put onto the page as usual.
In reality, if your data is more complex than this, it may be better to send the data to the script via the post method.