Widget Development – Displaying dropdown content

$instance['page_id'] should be the post ID of the selected post. So …

function widget($args, $instance)
    $post = get_post( $instance['page_id'] );
    echo $post->post_content; // you should add the common filters here
    var_dump( $post ); // more data

… should be a good start.

To save the last value in the configuration form use selected():

foreach( $posts as $post )
    setup_postdata($post); ?>
        <option value="<? echo $post->ID; ?>" <?php 
        selected( $instance['page_id'], $post->ID ); 
        ?>><?php the_title(); ?></option>