Obvioulsy you echo image_first value, but you need to echo result of REPLACE. You can modify query as:
mysqli_query($conn, "SELECT image_first, REPLACE(image_first,'/home/erik/','') as new_img FROM reviews_media WHERE review_id = $id");
See, I added an alias to result of REPLACE function. Now you can echo something like:
echo $row['new_img'];
As you don't do anything to result of REPLACE in a query, you can also simplify it and do a replace with php:
mysqli_query($conn, "SELECT image_first FROM reviews_media WHERE review_id = $id");
// fetching results
echo str_replace('/home/erik/', '', $row['image_first']);