How can I get the author description’s excerpt?

There is none. You have to implement your own custom function to trim any content you wish. For example, this function will trim your content based on words:

function my_custom_excerpt ( $content, $limit = 20, $more="..." ){                      
    return $data = wp_trim_words( strip_tags( $content ), $limit, $more );
}

Or this one will trim it based on characters:

function my_custom_excerpt( $excerpt, $limit = 20 ) {
    $charlength++;
    if ( mb_strlen( $excerpt ) > $charlength ) {
        $subex = mb_substr( $excerpt, 0, $charlength - 5 );
        $exwords = explode( ' ', $subex );
        $excut = - ( mb_strlen( $exwords[ count( $exwords ) - 1 ] ) );
        if ( $excut < 0 ) {
            $output = mb_substr( $subex, 0, $excut );
        } else {
            $output = $subex;
        }
        $output .= ' ...';
        return $output;
    } else {
        return $excerpt;
    }
}

Now you can use:

<p><?php echo my_custom_excerpt( get_the_author_meta('description') , 20 ); ?></p>

Which will trim the author’s description by 20 words/characters, based on your function.