How to work async in jQuery ajax | Does it have something to do with preventing other events on the page from firing?

Yes, async to false means that the statement you are calling has to complete before the next statements of your function can be called. If you set async: true then that statement will begin it's execution the same time and the next statement will be called regardless of whether the async statement has completed yet.

jQuery('body #pricegroup-products-container').on('click', '.saveimage' , function (e) {
        var returnHtml = '';
        jQuery.ajax({
            type: "POST",
            async: false,
            cache: false,
            url: url,
            data: {id: '12345'},
            dataType: "html",
            success: function (response) {
                returnHtml = response;
            }
        });
        alert(returnHtml);
    });