How to replace text in hompage only

the_content(Read more ...') echoes the stuff straight out so you can’t modify it with str_replace after the fact.

If you wanna use str_replace you need to do something along the lines of

   $content = get_the_content( 'Read more ...' );
   $content = apply_filters( 'the_content', $content );
   $content = str_replace( ']]>', ']]>', $content );
   $content = str_replace( 'xxx', 'yyy', $content );
   echo $content;

and replace xxx and yyy with the real stuff ofc.