Relevanssi can highlight search terms on the posts. But how about scrolling the page to show the location where the search results are? That is also possible.

It requires a bit of JavaScript on the post page.

This script needs to run on the post page:

<script>
jQuery(document).ready(function($) {
	$.extend($.expr[":"], {
		"containsNC": function(elem, i, match, array) {
			return (elem.textContent || elem.innerText || "").toLowerCase().indexOf((match[3] || "").toLowerCase()) >= 0;
		}
	});

	var mySearchString = getParams("highlight");
	var offsetToWord = $("p:containsNC('" +mySearchString + "'):first").offset().top;
	
	$("html, body").animate({ scrollTop: offsetToWord }, 0);
	
	function getParams(param) {
		var vars = {};
		window.location.href.replace( location.hash, '' ).replace( 
			/[?&]+([^=&]+)=?([^&]*)?/gi, // regexp
			function( m, key, value ) { // callback
				vars[key] = value !== undefined ? value : '';
			}
		);
		return vars[param];
	}
});
</script>

The easiest way is to just add this to the single post template, but the right way is to use wp_enqueue_script(). Now the post page will automatically scroll to the location of the search term, taken from the “highlight” query parameter. The scroll is instant; if you want to make it slower, increase the 0 in $("html, body").animate({ scrollTop: offsetToWord }, 0); to 1000 or 2000 to get a slower animation.

Now all you need is the search term in the highlight parameter. On your search results template, you likely have something like this:

<a href="<?php the_permalink(); ?>"><!--?php _e( 'Read More', 'total' ); ?--></a>

Change it to this to append the search query to the URL:

<a href="<?php if (function_exists('relevanssi_the_permalink')) { relevanssi_the_permalink(); } else { the_permalink(); } ?>"><!--?php _e( 'Read More', 'total' ); ?--></a>

This is a fairly brutal method and probably lacks all sorts of necessary finesse, but it’s a start. Note that this only works when the whole search query is found somewhere on the page; it looks for the whole phrase, not individual words.

Another method

For a simpler and more reliable approach, make sure the in-document highlighting is enabled and set it to use something distinct, like that <mark> tag or a specific CSS classname. Then you can use this script to jump to the first occurrance of the highlighting:

<script>
jQuery(document).ready(function($) {
    var offset = $('mark').first().offset().top // for <mark> tags
    var offset = $('.classname').first().offset().top // for CSS class name
    $("html, body").animate({ scrollTop: offset }, 100);
})
</script>