Insert Sponsor logo in custom own theme

That’s a shortcode, it’s meant to go in content. If you want to output it in a template but don’t know the underlying function then you can use do_shortcode():

<?php echo do_shortcode( '[sponsor_level category="all" orderby="menu_order" columns="4" image="yes" title="no" link="yes" bio="yes" show="9999999"]' ); ?>