wp_dropdown_categories() function output a select box. You can wrap the output within a form with form action to the same page and upon user submitting the form, you can get the category id value and query for the posts within that category. Or you can use jQuery so that you don’t need any form at all.
Codex has nice example for both of the case.
If you want to post the form to current page, you can use that page url as the form action like this
<li id="categories">
<h2><?php _e( 'Categories:' ); ?></h2>
<form id="category-select" class="category-select" action="<?php echo get_permalink($page_id); ?>" method="get">
<!-- $page_id is the current page id -->
<?php wp_dropdown_categories( 'show_count=1&hierarchical=1&name=category_select' ); ?>
<input type="submit" name="submit" value="view" />
</form>
</li>
You can use the returned value like this
$paged = ( get_query_var( 'paged' ) ) ? get_query_var( 'paged' ) : 1;
$cat = ( isset( $_GET['category_select'] ) ) ? $_GET['category_select'] : 1;
$args = array (
'cat' => $cat,
'posts_per_page' => 10,
'paged' => $paged
);
$query = new WP_Query( $args );