There are a lot of options here I believe.
Here is my solution(s): You could use search_array() for this. The search array returns false if no other needle is found in_array in an array. If another word is found it returns the key. 
Depending of your needs, you could use one of these options below.
//Option 1
//Words that actually appear more than once... 
$new_arr = array();
foreach($exp as $key=>$e) {
    //Must be this word only (therefore the true-statement
    $search = array_search($e, $exp, true); 
    if ($search !== false && $search != $key) {        
        $new_arr[] = $e;
    }
}
//Option 2
//
//Your question was not totally clear so I add this code as well
//Words with asterixes before and after that appear more than once
$new_arr = array();
foreach($exp as $key=>$e) {
    //Two asterixes at the beginning of the sting and two at the end
    //strtolower sets **Technology** and **technology** as a duplicate of word
    if (substr($e,0,2) == "**" && substr($e,-2,2) == "**") { 
        $search = array_search(strtolower($e), $exp);
        if ($search !== false && $search != $key) {        
            $new_arr[] = $e;
        }
    }
}
for($j = 0; $j < count($new_arr); $j++){
    $this->db->query("INSERT INTO news (news_id, news_content) 
    VALUES ('$id', $new_arr[$j])");
}
As someone mentioned in a comment you should prevent SQL injections by typing that way in the INSERT-statement (and you should), but the question was mainly about finding duplicates in a string in do something  with them so therefore I won't go further with that comment.
The result array $new_arr would like: (option 1)
array (size=9)
  0 => string 'the' (length=3)
  1 => string 'the' (length=3)
  2 => string '**food**' (length=8)
  3 => string 'to' (length=2)
  4 => string 'the' (length=3)
  5 => string '**impact**' (length=10)
  6 => string 'have' (length=4)
  7 => string 'on' (length=2)
  8 => string 'the' (length=3)
The reason why Technology and technology is not the same because that its an uppercase T in one of the words.
The result array $new_arr would like: (option 2)
array (size=3)
  0 => string '**food**' (length=8)
  1 => string '**Technology**' (length=14)
  2 => string '**impact**' (length=10)