Get all the posts where meta field with multiple choice has several values checked

did you try to call like this:

$args = array(
  'post_type'  => 'product',
  'meta_query' => array(
    'relation' => 'OR',
    array(
        'key'     => 'spec',
        'value'   => 'choice2',
        'compare' => 'LIKE',
    ),
    array(
        'key'     => 'spec',
        'value'   => 'choice3',
        'compare' => 'LIKE',
    ),
  ),
);
$query = new WP_Query( $args );

example taken straigh from the codex