You’re missing <?php and ?>
<?php
if( in_category( 'movies' ) ) { // this (end php, start html) -> ?>
<span style="font-size:16px">
<strong>Movie:</strong> <?php the_field( 'movie_name' ); ?> (<?php the_field( 'movie_year' ); ?>)<br/>
<strong>IMDb:</strong> <?php the_field( 'imdb_link' ); ?><br/>
<strong>Rotten Tomatoes: <?php the_field( 'rt_link' ); ?> <br/><br/>
</span>
<br/><br/> <?php // <- this (end html, start php)
} ?>